Pharmacokinetics Simulations for Studying Correlates of Prevention Efficacy of Passive HIV-1 Antibody Prophylaxis in the Antibody Mediated Prevention (AMP) Study
Lily Zhang Peter B. Gilbert Edmund Capparelli Yunda Huang
q-bio.QM stat.AP

[
    M P Gochan^1, Hua Li^1, K S Bedell^1
    
========================================





Abstract

A key objective in two phase 2b AMP clinical trials of VRC01 is to evaluate whether drug concentration over time, as estimated by non-linear mixed effects pharmacokinetics (PK) models, is associated with HIV infection rate. We conducted a simulation study of marker sampling designs, and evaluated the effect of study adherence and sub-cohort sample size 
on PK model estimates in multiple-dose studies. With m=120, even under low adherence (about half of study visits missing per participant), reasonably unbiased and consistent estimates of most fixed and random effect terms were obtained. Coarsened marker sampling schedules were also studied. 




keywords: multiple-dose; NONMEM; population PK analysis; simulation-based sampling design; study adherence; two-compartment PK model.



§ INTRODUCTION

VRC01 is a human IgG1 broadly neutralizing monoclonal antibody (mAb) directed against the CD4-binding site of HIV-1 (<cit.>). 
In 2016, the HIV Vaccine Trials Network and HIV Prevention Trials Network launched the VRC01 Antibody Mediated Prevention (AMP) study, the first efficacy study of a broadly neutralizing anti-HIV antibody for prevention of HIV infection. The study is being conducted in two harmonized trials  in two cohorts: 2700 HIV-uninfected men and transgender persons in the Americas and Switzerland; and 1500 HIV-uninfected sexually active women in sub-Saharan Africa (<cit.>). Within each cohort, AMP participants are randomized to receive 10 IV infusions of 10 mg/kg VRC01, 30 mg/kg VRC01, or placebo every 8 weeks through 72 weeks. Besides the evaluation of prevention efficacy of VRC01, an important secondary objective of AMP is to assess VRC01 mAb markers, e.g. serum mAb concentrations over time, as correlates of protection (CoP) against HIV-1 infection. This knowledge is anticipated to help guide further development of mAbs and to provide benchmarks for HIV-1 vaccine development.


In the AMP correlates study, a case-control sampling design is used. Markers are measured from primary endpoint HIV-1 infected cases in the mAb groups and from a random sample of control participants from each mAb group who remain HIV-1 negative  until at least the Week 80 study visit  (<cit.>). While complete pharmacokinetics (PK) information (e.g. concentration at any given time) is desirable especially in the study of time-dependent correlates, it is infeasible to sample continuously or on an intensive time scale. Therefore, the full trajectory of VRC01 concentrations over time needs to be estimated based on the concentration data collected at selected time-points for the correlates study. As specified in the AMP protocol, 4-weekly blood samples were draw for possible marker measurements at up to 22 visits prior to Week 80  (i.e., 4 weeks and 8 weeks [trough] after each 8-weekly infusion), and at the visit scheduled 5 days after the second infusion (Figure 1). Of note, in the real trials these sampling times are subject to variations as a result of imperfect study adherence due to possible missed visits, permanent infusion discontinuations and study dropout, and actual visit windows around each target visit date (Supplementary materials). Therefore, besides different marker sampling designs, we also consider different levels of study adherence and their impact on the outcomes in the simulations described herein.


In this paper, we focus on the modeling of VRC01 concentrations among hypothetical HIV-uninfected control participants in AMP, defined as participants assigned to a VRC01 group who reach the Week 80 visit HIV-1 negative. Because all AMP participants acquiring HIV-1 infection during the study are sampled for measuring VRC01 concentrations at all available sampling time-points, the relevant sampling question is how many control participants and what time-points to sample in the AMP correlates study. Besides a range of sample sizes of control participants, we consider time-point sampling  according to: 1) the complete schedule of all 22 possible visits; and 2) coarsened schedules of only a subset of visits. In addition, because mAb serum concentrations often change non-linearly over time with possible individual-to-individual variability, population PK (popPK) analysis based on non-linear mixed effects models is used to estimate population and individual PK parameters, as well as to estimate individual-level serum concentrations over time (with associated estimation uncertainties) that can be assessed as potential correlates.


The outline of this paper is as follows. In Section 2, we introduce the Master PK model that is used for the simulation of AMP participants' serum concentration data; the simulation set-up, including different marker sampling designs for the AMP control cohort; and the method for estimating PK model parameters based on the simulated datasets. In Section 3, we compare the performance of different complete and coarsened schedule marker sampling designs in terms of the accuracy and precision of the PK parameter estimates. We draw some conclusions and recommendations in Section 4. 



§ METHODS



 §.§ Master PK model

A two-compartment model is deemed fitting to describe the PK of VRC01, since this mAb has limited distribution volume, slow clearance, and hence a long half-life (∼15 days). Non-linear mixed effects models are used to describe individual-to-individual processes of drug absorption, distribution, and elimination and how these vary across individuals of different characteristics (e.g. weight, sex, or age). We use the popPK model that best describes the serum concentration data collected in a phase 1 study of VRC01 <cit.> as the prototype Master model for the simulation of concentration data from AMP participants. In the following, we briefly review the Master popPK model with two levels: the individual-level PK (or structure) model, which describes the intra-individual patterns of serum concentration over time through the processes of drug absorption, distribution, and elimination; and the population-level (or variability) model, which describes the inter-individual variability of the processes.



  §.§.§ Individual-level PK model

 et S(t_ij) denote the VRC01 serum concentration at the j^th measurement for subject i. In the Master individual-level PK model, it is defined as 


    S(t_ij)= f(t_ij,D_i,β_i)*(1+e1_ij) + e2_ij, i=1, …, m,  j=1,…,m_i
 where f denotes a two-compartment model with dose D_i, β_i=(CL_i,V_c_i,V_p_i,Q_i) denotes parameters of f specific to individual i, and the proportional error and additive constant error terms e1_ij and e2_ij satisfy, respectively, (e1_ij|D_i,β_i)=0, (e1_ij|D_i,β_i)=σ_1^2 and (e2_ij|D_i,β_i)=0, (e2_ij|D_i,β_i)=σ_2^2. There are different ways of parameterizing a compartment model. For their direct physiology significance, CL: clearance rate (L/day), V_c: volume of distribution in the central compartment (L), V_p: volume of distribution in the peripheral compartment (L), and Q: inter-compartmental clearance (L/day) are often a set of parameters used to describe PK processes underlying the observed concentration profiles for a given subject under a two-compartment model. 


Specifically, after a single IV infusion, the serum drug concentration at time t (relative to the end of infusion) can be  expressed as the sum of two exponential processes – distribution and elimination (i.e. a two-compartment model):


    S(t)     =    D/T_inf[A/α(1-e^-α T_inf)e^-α t + B/β(1-e^-β T_inf)e^-β t]

where D= IV dose amount (mg); T_inf= duration of infusion; α and β are rate constants (slopes) for the distribution phase and elimination phase, respectively (α > β by definition); and A and B are intercepts on the y axis for each exponential segment of the curve. Of note, when α T_inf→ 0  and β T_inf→ 0 [as is the case for IV bolus administrations and is approximately the case for IV VRC01, where the infusion time is brief (∼30 minutes) relative to the half-life], S(t)=D[Ae^-α t + Be^-β t] because lim_h→ 0e^h-1/h=1. Let k_12=Q/V_c and k_21=Q/V_p represent the first-order rate transfer constants (1/day) for the movement of drug between the central and peripheral compartments. Mathematically, α and β relate to the rate constants k_12 and k_21 as α+β  =  k_12 + k_21 + k and 
αβ  =  k_21k, where k= CL/V_c= αβ(A+B)/Aβ+Bα is the rate of drug elimination. In addition,
A =  α-k_21/V_c(α-β) and 
B  =  k_21-β/V_c(α-β).
After multiple doses, serum drug concentration at time t after k doses D_j  (j=1,.…, k) given at time t_D_j (t ≥ t_D_k) can then be expressed as written below, based on the superimposed assumption that the PK of the drug after a single dose are not altered after taking multiple doses: 


    S(t)    =   ∑_j=1^kD_j(Ae^-α(t-t_D_j) + Be^-β(t-t_D_j))




  §.§.§ Population PK model

Let β_i = d(a_i, β, b_i),  i=1, …, m
where d denotes a p-dimensional function, a_i denotes a vector of characteristics for subject i that contribute to explaining inter-individual variability, β denotes fixed effects, and b_i denotes random effects. This equation characterizes how elements of β_i vary across individuals due to systematic association with a_i (modeled via β) and unexplained variation in the population represented by b_i. The usual assumptions (b_i|a_i) = (b_i) = 0 and (b_i|a_i)= (bi) = Ω are implied. 


In the Master population-level PK model, all four individual-level PK parameters are modeled as log-normally distributed with exponential inter-individual variability (IIV) random effects. Let CL_i, V_c_i, Q_i, and V_p_i denote individual-level estimated CL, V_c, Q, and V_p, respectively, for individual i. Covariates a_i include only BW_i, body weight of subject i, which has an influence on CL_i and V_c_i via an exponential model as follows: 



    CL_i    =   β_CL * exp( β_BW.CL * (BW_i-74.5)) * exp(b_CL_i)
    
    V_c_i   =   β_V_c * exp( β_BW.V_c * (BW_i-74.5)) * exp(b_V_c_i)
    
    Q_i    =   β_Q * exp(b_Q_i)
    
    V_p_i   =   β_V_p * exp(b_V_p_i)





 §.§ Simulation set-up



  §.§.§ Study adherence

Each participant's infusion and post-infusion visit schedule is simulated according to the AMP protocol specification (Supplementary Materials). Briefly, participants receive 10 IV infusions of VRC01 (or placebo). Each infusion visit has a visit window of -7 to +48 days around the scheduled 8-weekly infusion visit target date. Participants' marker measurements occur at each infusion visit, 4 weeks (visit window: -7 to +7 days) after each infusion, and 5 days (-2 to +2 days) after the second infusion. In the following simulations, for infusion visits, we assume that 80% of the attended visits occur uniformly during the target window of -7 to +7 days and 20% during the allowable window of +7 to +48 days. For post-infusion visits that occur between infusion visits, we assume all attended visits occur uniformly during the specified window. 


Overall, we consider four study adherence patterns defined by the combinations of four missing data probabilities: probability of an independently missed single infusion (p_1), probability of an independently missed post-infusion visit (p_2), cumulative probability of permanent infusion discontinuation (r_1), and annual dropout rate (r_2). The four study adherence patterns in terms of (p_1, p_2, r_1, r_2) are: perfect adherence = (0%, 0%, 0%, 0%), high adherence = (5%, 10%, 10%, 10%), medium adherence = (10%, 15%, 15%, 15%), and low adherence = (15%, 20%, 20%, 20%). At 1 year after trial initiation, the adherence is very high in the ongoing AMP study with rates of  (2%, 3%, 5%, 5%). Nevertheless, a wider range of adherence levels are considered in this paper for interpolation purposes and for investigating the robustness of popPK modeling against missing data, relevant if adherence declines in AMP or for future mAb studies. 



  §.§.§ PK model parameter values

Once participants' infusion and post-infusion visits are simulated according to the study adherence patterns described above, the Master popPK model described in section 2.1 is used to simulate VRC01 serum concentration at attended study visits for participants of a given body weight in each VRC01 dose group (10 mg/Kg and 30 mg/Kg). The PK parameter values used for the simulation are listed in Table <ref>, where low covariances between random effects are fixed at zero to increase model stability. 



  §.§.§ Computation software

R version 3.3.1 <cit.> was used for the simulation of participants' characteristics and study visit data. The NONMEM software system (Version 7.3, ICON Development Solutions) was used for the simulation and modeling of concentration data. Parallel computing on eight central processing units was employed to speed up computation time via NONMEM. 



  §.§.§ Marker sampling design

We consider two types of marker sampling designs: complete schedule and coarsened schedule (Figure 2). The former design includes concentration data at all time-points of visit attendance from each participant, whereas the latter includes concentration data only at a subset of time-points. Because a total of 61 VRC01 HIV-infected cases are expected at 60% prevention efficacy for both dose groups pooled over both AMP trials, for the complete schedule design, we consider sample sizes of m = 30, 60, 120, and 240 HIV-1 uninfected controls. The latter three sample sizes correspond to the numbers of expected controls in the case-control cohort with a 1:1, 1:2, and 1:4 case:control ratio, respectively, whereas m = 30 serves as a reference and represents the 1:1 case:control ratio for a single AMP trial. For the coarsened schedule design, we consider only one sample size of m=240, but 3 coarsened schedules that sample roughly half of the complete schedule time-points per participant. 


  * First half: the first 11 time-points (excluding time 0) out of the total 22 complete schedule time-points are sampled. These time-points include, for every individual the 4-week and 8-week post-infusion time-points (trough) after the first 5 infusions, in addition to the 5-day time-point after the 2nd infusion. 

  * Mixed half: time-points after every other infusion are sampled. In addition to the 5-day time-point after the 2^nd infusion for every individual, 4-week and 8-week time-points (trough) after the five odd number (1^st, 3^rd, 5^th, 7^th, and 9^th) infusions are included for one half of the subjects (m=120), and 4-week and 8-week post-infusion time-points after the five even number (2^nd, 4^th, 6^th, 8^th, and 10^th) infusions are included for the other half of the subjects (m=120).

  * Trough only: trough time-points are sampled. These include, for every individual the 8-week time-points (trough) after each of the 10 infusions, in addition to the 5-day time-point after the 2nd infusion.

Note that for all 3 coarsened schedules, the 5-day time-point after the 2^nd infusion is always included because this  time-point is the only one scheduled within a few days of an infusion for the estimation of V_c. These 3 coarsened schedules are compared to the complete schedule with m=120, since the same number of 4-week and 8-week post infusion observations are made. 



  §.§.§ Simulation steps

For the complete schedule marker sampling design, we consider a total of 16 scenarios representing combinations of 4 study adherence patterns (perfect, high, medium, and low) and 4 sample sizes (m=30, 60, 120, and 240). For the coarsened schedule marker design, we consider a total of 12 scenarios representing the combinations of 4 study adherence patterns and 3 coarsened schedules (First half, Mixed half, and Trough only). The complete schedule datasets with m=240 are first simulated and the coarsened schedule datasets are extracted from them given the specific design. For each scenario, 1000 datasets are simulated containing participants' demographic information (body weight and sex), study information (infusion and post-infusion visit time), and serum concentration at attended visits. Depending on the research interest, concentration values on consecutive days or flexible grid can also be simulated to evaluate the modeling/prediction of simulated concentration using sparse data. Specifically, each dataset is simulated following the steps described below:


  Step 1: Simulate participants' characteristics representing the two AMP study cohorts and their corresponding VRC01 dose amounts in the low and high dose groups. 


  a.Simulate 1:1 male:female sex ratio.

  b.For females and males, respectively, randomly sample participants' body weight with replacement from females in the Phambili trial in South Africa <cit.>, and from males in the Step study in the Americas <cit.>. Participants' body weights are assumed to be constant throughout the trial. 

  c. Within each sex, participants are randomly assigned to the 10 mg/Kg and 30 mg/Kg dose groups at a 1:1 ratio. Each participant's VRC01 dose amount is determined as the product of his/her body weight and dose level.




  Step 2: Simulate participants' 8-weekly infusion visits.


  a. Simulate the attendance (yes or no) of each of the 10 infusion visits using a Bernoulli probability of p_1.

  b. If attendance is `yes' for a given infusion visit, simulate the infusion visit time according to the AMP protocol-specified schedules and visit windows, assuming the probability of attending the visit within the target window (typically -7 to +7 days) is 80% and the probability of attending the visit outside the target window but within the study allowable window (typically -7 to +48 days) is 20%. The infusion time follows a piece-wise uniform distribution within and outside the target window.




  Step 3: Simulate participants' post-infusion visits: baseline, 4-weekly, and 5-day post-2^nd infusion.


  a. Simulate post-infusion visit attendance (yes or no) according to prior infusion attendance. If the prior infusion is administered, simulate the visit attendance using a Bernoulli probability of p_2. If the prior infusion is missed, then the 5-day post-2^nd infusion visit (if applicable) and the following 4-week post-infusion visits are all considered missed. If the last infusion (#10) is missed, then the 4-week post-infusion visit is also missed. However, the 8-week post infusion visit could still be scheduled.

  b. If post-infusion attendance is `yes', simulate post-infusion visit time accounting for prior infusion visit time, according to protocol-specified schedules/windows, similar to the procedure described in Step 2b. 




  Step 4: Modify infusion attendance and post-infusion visits accounting for permanent infusion discontinuation.


  * Simulate time to permanent infusion discontinuation (due to reasons other than HIV infection) using a random exponent rate r_1.

  * Modify infusion attendance and post-infusion visits for those who discontinue infusion permanently according to a different protocol-specified visit schedule/window (Supplementary Materials).




  Step 5: Modify infusion attendance and post-infusion visits accounting for dropout (study termination).


  * Simulate time to dropout using a random exponent rate r_2.

  * Censor all previously simulated infusion and post-infusion visits at dropout time.




  Step 6: Simulate participants' concentrations according to the Master PK model, with the covariate information, dose amount, infusion, and visit schedules as in the previous steps. 


  (a) Simulate S(t_ij) according to the final PK model by setting b_i=0 and ϵ_ij=0.

  (b) Add the above value to a mean-zero normally distributed b_i and ϵ_ij, according to the variance estimates from the final PK model.





 §.§ Estimation Method
 
Parameter estimation of the popPK model is based on minimizing the objective function value using maximum likelihood estimation. A marginal likelihood of the observed data is calculated based on both the influence of the fixed effect and the random effect. Different estimation methods for non-linear mixed effects models have been extensively discussed by other authors (e.g. <cit.>). In this paper, due to the sparseness of the simulated data, the Markov chain Monte Carlo stochastic approximation expectation-maximization (SAEM) method <cit.> is applied to the modeling of the simulated time-concentration data according to the Master PK model. The true values of each PK parameters as specified in Table 1 are used as initial values. 



§ RESULTS

For each scenario of the complete schedule and coarsened schedule designs, we report 

    
  1) % datasets `converged'. Due to the Monte Carlo nature of the SAEM method, convergence testing is not formally done.  Completed runs are counted as convergence successes.
    
  2) For each fixed and random effect: 6 fixed-effect terms, 6 random-effect terms, and 2 residual error terms, among the B converged models, 
    
    
  * relative bias, RBias =1/K∑^B_k=1(β̂_k - β/β)*100, 
    
  * relative root mean squared error, RRMSE =√(1/K∑^B_k=1(β̂_k - β/β)^2)*100, and 
    
  * coverage probability, CP=proportion of datasets with 95% confidence intervals including the true value of the parameter β. 


  3) Shrinkage estimate for each random-effect terms, i.e. one minus the ratio of the standard deviation of the individual-level estimate and the estimated variability for the population estimate. 




 §.§ Master popPK model

As an illustration, Figure 2 displays two expected population-level time-concentration curves for individuals who are perfectly adherent to the 8-weekly infusion schedule, one for the 10 mg/Kg dose and one for the 30 mg/Kg dose, based on the master popPK model with E(b_i)=0, E(e1)=0, and E(e2)=0. The concentrations at each time-point are simulated based on a body weight of 74.5 Kg and with the PK parameter values given in Table 1.

In addition, a random set of the simulated individual-level concentration curves under low study adherence are displayed in Figure 3. For example, individual #2 in the low dose group (left panel) stayed in the study for follow up but discontinued infusions after the first infusion, whereas individuals #9 and #26 in the high dose group (right panel) dropped out of the study right after the first post-infusion visit.



 §.§ Model fitting



  §.§.§ Complete schedule marker sampling designs

Using the SAEM estimation method, almost all models using datasets under the complete schedule designs converged to obtain final PK parameter estimates for m=120 and m=240, whereas a relative low convergence was observed for datasets with m=30 and m=60 (Figure 4). This result suggests that a minimal sample size of m=60 is recommended for a stable PK model fitting under the described schedule. This suggestion is further confirmed when the accuracy and precision of the fixed-effect estimates are examined (Figures 5 &6, Supplementary Materials: Table 1). Reasonable levels of bias and precision with significant improvements over the small sample sizes are observed for m=120. On the other hand, m=240 provides relatively marginal improvements compared to m=120 except for β_BW.CL, which characterizes the effect of each individual's body weight on CL and requires data from a sufficient number of independent subjects for an accurate and precise estimation.


Due to the unstable estimation for n=30, we restrict the evaluation of the estimation of random effects to larger sample sizes (Supplementary Materials: Figures S1 & S2). In general, regardless of the sample size, the random effect of the PK parameter Q (inter-compartmental clearance rate) is poorly estimated with high shrinkage (Supplementary Materials: Figure S3). This poor estimation is due to the sparsity of data closer to infusion, with only one 5-day post infusion time-point, and the low inter-individual variability in Q. On the other hand, the estimation of CL and V_p seems reasonable, with shrinkage generally below 20-30%. The proportional error term, σ_1^2 is also reasonably estimated with RRMSE < 10% under all scenarios. The estimation of the additive error term, σ_2^2 is relatively poorer possibly due to the sparsity of data around the assay limit of detection. 



  §.§.§ Coarsened schedule marker sampling designs

Figure 7 displays the distribution of the total number of observations (5 days after the 2^nd infusion, 4 weeks and 8 weeks after each infusion) based on the 1000 simulated datasets under each of the 12 coarsened schedule scenarios with m=240, along with the complete schedule design with m=120. Because about half of the complete schedule time-points are sampled in the coarsened schedules, the total expected number of 4-week and 8-week post infusion observations are the same across the 4 designs. This feature allows a fair comparison across the designs. Meanwhile, the number of 5-day post second infusion observations is doubled in the coarsened schedule designs, because the 5-day post 2^nd infusion time-point is the only time-point proximal to an infusion. Hence, this time-point is always sampled from every individual under both the complete and coarsened schedule designs. This consistency allows the assessment of the impact of 5-day post infusion observations on the estimation of various PK parameters.


Results showing the accuracy and precision of the fixed-effect estimates under each coarsened schedule design with m=240 are displayed in Figures 8, 9 and Supplemental Materials (Table 2). The complete schedule design with m=120 is included as a reference for comparison purposes. In general, for the estimation of fixed effects, the complete schedule design with half of the sample size provides more accurate but less precise estimates, except that more accurate estimates are obtained for β_V_c and β_BW.CL under the `First half' and `Mixed half' coarsened schedule designs. This is likely due to the fact that having more 5-day post infusion observations in the coarsened schedule designs helps improve the estimation of β_V_c, which requires data proximal to infusion for an accurate estimation, and having more independent individuals helps improve the estimation of the covariate effect. On the other hand, the accuracy of estimates under the coarsened schedule designs are more impacted by study adherence due to the sparser time-points compared to the complete schedule design. Among the 3 coarsened schedule designs, the `First half' and `Mixed half' designs have very similar performance and are generally superior to the `Trough only' design, especially for the estimation of β_V_c. Similar patterns are observed in the estimation of random effects (Supplemental Materials: Figures S4-S6). Poor estimation of the random effect of inter-compartment clearance (Q) and additive residual error are observed for all designs for reasons stated above.



§ CONCLUSIONS

PopPK analysis is known to be suitable for datasets consisting of a few data points per individual over the course of product administration(s) from many individuals, in order to estimate popPK parameters adjusting for variability among individuals. In this paper, we investigated how the accuracy and precision of the estimated population parameters (fixed effects) and variabilities among individuals (random effects) are influenced by the number of individuals and by the number and type (i.e. time-point) of observations per individual.


In the context of the AMP study, where participants receive ten 8-weekly IV infusions of VRC01, we considered complete schedule marker sampling designs where approximately 4-weekly observations from up to 22 time-points over the course of 80 weeks are included in the popPK modeling, with 4 different levels of study adherence (perfect, high, medium, and low) and 4 different sample sizes (m= 30, 60, 120, and 240). We found that a sample size of 120 or higher could render reasonably unbiased and consistent estimates of most fixed and random effect terms. The central volume parameter V_c is the most challenging fixed effect parameter to estimate due to the lack of concentration data proximal to infusion as specified in the AMP protocol.
 

We also considered coarsened schedule marker sampling designs with m=240, where the first half (`First half'), alternate (`Mixed half'), or `Trough only' time-points are included in the popPK modeling. These designs often provide less accurate but more precise estimates of various popPK parameters than the complete schedule design with m=120. In terms of overall estimation performance as measured by RRMSE, the `First half' and `Mixed half' designs render similar performance, but are generally superior to the `Trough only' design and the complete schedule design. We note that the `First half' design is less subject to missing data, but provides limited data in the assessment of the steady state and the effect of a higher number of repeated doses. On the other hand, the `Mixed half' design is more subject to missing data due to infusion discontinuation and study drop out as the study progresses. Based on these simulation results, we favor using the `Mixed half' design for the AMP case-control study, given that it provides the best overall accuracy and precision for various PK parameter estimates in studies of high adherence like the current AMP study. In addition, the `Mixed half' design allows the assessment of concentrations after any of the ten infusions (as opposed to only the first five infusions in the `First half' design); these data may be helpful in the analysis and interpretation of other study endpoints including long-term safety and anti-drug activity that may occur later in the study. If adherence declines in AMP, the advantages of the coarsened schedule designs will diminish and the full schedule design may be considered. 


In summary, this paper provides a simulation-based framework to evaluate sampling designs of multiple-dose PK studies using a stochastic process for participants' characteristics (e.g. sex and body weight) and infusion/measurement time-points. It also provides a simulator for studying statistical methods for assessing prevention efficacy and correlates of prevention efficacy. This simulator not only accounts for participant characteristics that influence PK processes, but also accounts for possible missed or terminated product administrations, protocol-specific study visits and visit windows, and potential drop out. Thus, this framework provides a realistic simulator of PK data for future studies of repeatedly-administered drugs. 



§ DISCLOSURE STATEMENT

No potential conflicts of interest were disclosed. 


§ FUNDING

This work was supported by the National Institute of Allergy and Infectious Diseases (NIAID) US. Public Health Service Grant UM1 AI068635 [HVTN SDMC FHCRC]. The content of this manuscript is solely the responsibility of the authors and does
not necessarily represent the official views of the National Institutes of Health. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.




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Johansson MA, Ueckert S, Plan E, et al. Evaluation of bias, precision, robustness and runtime for estimation methods in NONMEM 7. J Pharmacokinet Pharmacodyn. 2014;41:223-38.


Delyon_AS1999 
Delyon B, Lavielle M, Moulines E.  Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 1999;27(1):94-128.

Kuhn_ESAIM2004 
Kuhn E, Lavielle M. Coupling a stochastic approximation version of EM with an MCMC procedure. ESAIM: P&S 2004;8:115-131.




















Figure Legends


Figure 1: AMP study schema and marker sampling designs. Panel A: a total of 2700 participants from the Americas and Switzerland trial and 1500 participants from the sub-Saharan Africa trial receive ten IV infusions (#1- #10) of VRC01 at 10 mg/Kg, 30 mg/Kg or placebo every 8-weekly at a 1:1:1 randomization ratio. Arrows indicate pre- and post-infusion time-points included in the complete schedule marker sampling design. Panel B: Orange circles indicate sampled time-points in each coarsened schedule marker sampling design. Additional specimen collection times at baseline (week 0) and after week 80 are not included in this figure because data at these time-points mainly contribute to assay control and safety monitoring, not in the PK modeling.
  



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Figure 2: Simulated time-concentration curves under the Master popPK model following ten 8-weekly IV infusions of VRC01 in the 10 mg/Kg and 30 mg/Kg dose groups with perfect study adherence. Solid lines are the medians; shaded areas are the 2.5% and 97.5% percentiles of the concentrations over 1000 simulated datasets. A body weight of 74.5 Kg is used in the simulations.




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Figure 3: Simulated time-concentration curves under the Master popPK model following ten 8-weekly IV infusions of VRC01 in the 10 mg/Kg and 30 mg/Kg dose groups with low study adherence. With low study adherence, probability of an independently missed single infusion (p_1), probability of an independently missed post-infusion visit (p_2), cumulative probability of permanent infusion discontinuation (r_1), and annual drop out rate (r_2) are assumed to be 15%, 20%, 20%, and 20%, respectively.
 



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Figure 4: Percent of models converged under the complete schedule designs with different levels of study adherence and sample sizes (m = 30, 60, 120, and 240.) The four study adherence patterns in terms of (p_1, p_2, r_1, r_2) are: Perfect = (0%, 0%, 0%, 0%), High = (5%, 10%, 10%, 10%), Medium = (10%, 15%, 15%, 15%), and Low = (15%, 20%, 20%, 20%), where p_1 indicates the probability of an independently missed single infusion, p_2 the probability of an independently missed post-infusion visit, r_1 the cumulative probability of permanent infusion discontinuation, and r_2 the annual drop out rate.
 



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Figure 5: Relative bias of each fixed-effect PK parameter estimate under complete schedule designs with different sample sizes (m = 30, 60, 120, and 240). β_CL, β_V_c, β_Q, and β_V_p indicate the fixed effect for CL, V_c, Q, and V_p, respectively. β_BW.CL and β_BW.V_c indicate the fixed effect of body weight influence on CL and V_c, respectively.




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Figure 6: Relative root mean squared errors of each fixed-effect PK parameter estimate under complete schedule designs with different sample sizes (m = 30, 60, 120, and 240). β_CL, β_V_c, β_Q, and β_V_p indicate the fixed effect for CL, V_c, Q, and V_p, respectively. β_BW.CL and β_BW.V_c indicate the fixed effect of body weight influence on CL and V_c, respectively.




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Figure 7: Distributions of the total numbers of 5-day post 2^nd infusion, 4-week post infusion, and 8-week post infusion observations under the complete schedule design with m = 120 and 3 coarsened schedule designs with m = 240. The `First half' design samples the first 11 time-points (excluding time 0) out of the total 22 complete schedule time-points. The `Mixed half' design samples time-points after every other infusion. The `Trough only' design samples only trough time-points. All 3 coarsened schedule designs always include the 5-day post 2^nd infusion time-point.




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Figure 8: Relative bias of each fixed-effect PK parameter estimate under coarsened schedule designs with m = 240 compared to the complete schedule design with m = 120. β_CL, β_V_c, β_Q, and β_V_p indicate the fixed effect for CL, V_c, Q, and V_p, respectively. β_BW.CL and β_BW.V_c indicate the fixed effect of body weight influence on CL and V_c, respectively. The `First half' design samples the first 11 time-points (excluding time 0) out of the total 22 complete schedule time-points. The `Mixed half' design samples time-points after every other infusion. The `Trough only' design samples only trough time-points. All 3 coarsened schedule designs always include the 5-day post 2^nd infusion time-point.




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Figure 9: Relative root mean squared errors of each fixed-effect PK parameter estimate under coarsened schedule designs with m = 240 compared to the complete schedule design with m = 120. β_CL, β_V_c, β_Q, and β_V_p indicate the fixed effect for CL, V_c, Q, and V_p, respectively. β_BW.CL and β_BW.V_c indicate the fixed effect of body weight influence on CL and V_c, respectively. The `First half' design samples the first 11 time-points (excluding time 0) out of the total 22 complete schedule time-points. The `Mixed half' design samples time-points after every other infusion. The `Trough only' design samples only trough time-points. All 3 coarsened schedule designs always include the 5-day post 2^nd infusion time-point.




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§ SUPPLEMENTAL MATERIALS









lclccc
Fixed effects PK parameter estimation accuracy and precision under complete schedule design scenarios. β_CL, β_V_c, β_Q, and β_V_p indicate the fixed effect for CL, V_c, Q, and V_p, respectively. β_BW.CL and β_BW.V_c indicate the fixed effect of body weight influence on CL and V_c, respectively. The four study adherence patterns in terms of (p_1, p_2, r_1, r_2) are: Perfect = (0%, 0%, 0%, 0%), High = (5%, 10%, 10%, 10%), Medium = (10%, 15%, 15%, 15%), and Low = (15%, 20%, 20%, 20%), where p_1= probability of an independently missed single infusion, p_2= probability of an independently missed post-infusion visit, r_1= cumulative probability of permanent infusion discontinuation, and r_2= annual drop out rate. RBias = average relative bias, RRMSE = average relative root mean squared error, and CP = coverage probability.



Parameter     Sample size     Study adherence     RBias (%)     RRMSE     CP (%)

[](continued)

Parameter     Sample size     Study adherence     Relative bias (%)     RRMSE     CP (%)

16*β_CL     4*30     Low     11.9	    26.07	    95.12

           Medium     7.91	   24.27	   97.54

           High     5.7	   16.84	   97.98

           Perfect    4.6	   18.78	   98.14

  2-6
      4*60     Low     0.43	   10.87	   98.85

           Medium    -0.56	   9.7	   99.29

           High    -0.45	   9.61	   98.79

           Perfect    -1.9	   8.93	   99.4

  2-6
      4*120     Low     0.88	   6.99	   98

           Medium    0.11	   6.37	   99

           High    0.18	   5.77	   98.9

           Perfect    -0.4	   5.17	   99.4

  2-6
        4*240     Low    1.36	   4.46	   98

           Medium    0.69	   4.07	   98.6

           High    0.4	   3.99	   99.1

           Perfect    -1.23	   3.77	   99.5

16*β_V_c     4*30     Low    35.46	   90.46	   94.7

           Medium    17.68	   76.53	   95.47

           High    12.83	   68.4	   96.79

           Perfect    4.05	   62.86	   96.83

  2-6
      4*60     Low    -1.09	   47.74	   97.49

           Medium    -6.91	   42.71	   97.35

           High    -5.62	   42.12	   97.37

           Perfect    -9.96	   38.26	   97.7

  2-6
      4*120     Low    3.03	   35.48	   98.2

           Medium    -1.39	   31.61	   99.1

           High    -2.83	   28.33	   98.6

           Perfect    -5.51	   25.32	   99.2

  2-6
        4*240     Low     3.39	   23.28	   99.1

           Medium    -1.14	   20.82	   99.4

           High     -2.3	   20.38	   98.9

           Perfect     -9.65	   19.31	   98.4

  16*β_Q     4*30     Low     9.63	   53.81	   97.77

           Medium    7.13	   50.32	   98.96

           High    6.75	   49.24	   99.52

           Perfect    2.93	   37.07	   99.45

 2-6
      4*60     Low     -0.06	   31.05	   99.69

           Medium    -1.15	   28.55	   99.69

           High     -0.43	   27.7	   100

           Perfect    -3.51	   24.35	   99.5

  2-6
      4*120     Low     3.65	   20.78	   99.5

           Medium     2.76	   20.01	   99.5

           High     2.92	   17.74	   99.5

           Perfect     1.37	   15.5	   99.8

  2-6
        4*240     Low     5.53	   14.18	   99.4

           Medium     4.13	   13.58	   99

           High     3.62	   12.69	   98.9

           Perfect    -0.83	   10.86	   99.4

  16*β_V_p     4*30     Low    3.86	   33.95	   99.58

           Medium    2.19	   34.38	   99.35

           High    0.6	   22.19	   99.64

           Perfect    1.08	   25.84	   99.78

  2-6
      4*60     Low    -1.44	   13.76	   99.79

           Medium     -1.57	   11.94	   99.49

           High    -1.44	   11.95	   99.9

           Perfect    -2.14	   10.7	   99.9

  2-6
      4*120     Low    0.64	   7.95	   99.3

           Medium    0.43	   7.27	   99

           High    0.81	   6.46	   98.8

           Perfect    0.27	   5.92	   99

  2-6
        4*240     Low    1.92	   4.77	   97.9

           Medium    1.64	   4.49	   98.2

           High    1.26	   4.45	   98.3

           Perfect    0.13	   3.95	   99

  16*β_BW.CL     4*30     Low    -12.15	   68.68	   97.21

           Medium    -17.1	   67.58	   96.89

           High    -17.78	   63.76	   98.1

           Perfect    -13.88	   60.21	   97.49

  2-6
      4*60     Low    -14.27	   50.84	   89.44

           Medium    -17.63	   50.81	   86.85

           High    -14.88	   46.7	   86.94

           Perfect    -14.61	   43.61	   87.88

  2-6
      4*120     Low    -9.1	   35.43	   86.17

           Medium    -7.43	   32.67	   87.2

           High    -4.72	   28.63	   90.4

           Perfect    -5.11	   25.98	   92.2

  2-6
        4*240     Low    -2.98	   20.97	   92.2

           Medium    -3.4	   21.33	   90.4

           High    -1.38	   16.18	   95.7

           Perfect    -2.24	   13.88	   96.2

  16*β_BW.V_c     4*30     Low    83.58	   175.16	   96.79

             Medium    99.46	   405.73	   97.67

           High    70.13	   195.68	   97.62

           Perfect    69.14	   152.71	   98.14

  2-6
      4*60     Low    28.14	   97.78	   97.8

           Medium    22.03	   89.87	   98.27

           High    20.75	   90.84	   98.18

           Perfect    12.48	   80.15	   97.7

  2-6
      4*120     Low    0.83	   70.7	   98

           Medium    -0.23	   66.91	   97.3

           High    5.68	   64.54	   96.5

           Perfect    1.52	   58.06	   97.4

  2-6
        4*240     Low    -1.85	   51.33	   96.7

           Medium    0.5	   50.1	   96.7

           High    2.36	   46.26	   95.5

           Perfect    1.55	   40.76	   95.6















lclccc
Fixed effects PK parameter estimation accuracy and precision under coarsened schedule design scenarios with n=240. β_CL, β_V_c, β_Q, and β_V_p indicate the fixed effect for CL, V_c, Q, and V_p, respectively. β_BW.CL and β_BW.V_c indicate the fixed effect of body weight influence on CL and V_c, respectively. The four study adherence patterns in terms of (p_1, p_2, r_1, r_2) are: Perfect = (0%, 0%, 0%, 0%), High = (5%, 10%, 10%, 10%), Medium = (10%, 15%, 15%, 15%), and Low = (15%, 20%, 20%, 20%), where p_1= probability of an independently missed single infusion, p_2= probability of an independently missed post-infusion visit, r_1= cumulative probability of permanent infusion discontinuation, and r_2= annual drop out rate. RBias = average relative bias, RRMSE = average relative root mean squared error, and CP = coverage probability.


Parameter     Sampling design     Study adherence     RBias (%)     RRMSE     CP (%)

[](continued)

Parameter     Sampling design     Study adherence     Relative bias (%)     RRMSE     CP (%)

16*β_CL     4*m=240 First half     Low     2.4	   5.24	   95.5

           Medium    1.83	   4.78	   96.9

           High    1.6	   4.76	   96.8

           Perfect    0.35	   4.05	   98.3
 
  2-6
      4*m=240 Mixed half     Low     2.23	   5.49	   96.5

           Medium    1.8	   5.13	   96.3

           High    1.85	   4.54	   97.3

           Perfect    0.43	   3.93	   98.9

  2-6
      4*m=240 Trough only     Low     4.25	   6.74	   92.4

           Medium    3.28	   6.02	   93.7

           High    2.75	   5.34	   95.2

           Perfect    1.8	   4.56	   97.2

  2-6
        4*m=120 Full     Low     0.88	   6.99	   98

           Medium    0.11	   6.37	   99

           High    0.18	   5.77	   98.9

           Perfect    -0.4	   5.17	   99.4
 
16*β_V_c     4*m=240 First half     Low    11.34	   29.68	   98.6

           Medium    7.35	   25.86	   98.7

           High    6.4	   25.73	   98.2

           Perfect    0.72	   20.42	   98.8

  2-6
      4*m=240 Mixed half     Low    11.51	   31.2	   98.6

           Medium    7.51	   27.46	   98.5

           High    7.21	   24.45	   99.1

           Perfect    0.71	   19.47	   98.9

  2-6
      4*m=240 Trough only     Low    21.97	   37.87	   97.4

           Medium    16.57	   33.08	   97.8

           High    13.65	   29.21	   98.4

           Perfect    10.85	   25.2	   99

  2-6
        4*m=120 Full     Low    3.03	   35.48	   98.2

           Medium    -1.39	   31.61	   99.1

           High    -2.83	   28.33	   98.6

           Perfect    -5.51	   25.32	   99.2

  16*β_Q     4*m=240 First half     Low     7.71	   15.94	   98.8

           Medium    6.69	   15.05	   98.5

           High    6.32	   14.64	   97

           Perfect    3.21	   12.31	   97.9

 2-6
      4*m=240 Mixed half     Low     6.79	   15.98	   99.4

           Medium    6.47	   15.99	   98

           High     7.05	   14.4	   98.1

           Perfect    3.4	   11.71	   98.4

  2-6
      4*m=240 Trough only     Low     -1.04	   13.55	   98.3

           Medium     0.99	   13.71	   98.3

           High     3.52	   13.98	   99

           Perfect     4.12	   13.12	   98.6

  2-6
        4*m=120 Full     Low     3.65	   20.78	   99.5

           Medium     2.76	   20.01	   99.5

           High     2.92	   17.74	   99.5

           Perfect     1.37	   15.5	   99.8

  16*β_V_p     4*m=240 First half     Low    1.66	   4.75	   97.5

           Medium    1.55	   4.54	   97.4

           High    1.17	   4.56	   97.7

           Perfect    0.68	   4.03	   98

  2-6
      4*m=240 Mixed half     Low    1.1	   4.86	   98.4

           Medium     1.24	   4.71	   97.4

           High    1.47	   4.3	   97.8

           Perfect    0.78	   4.13	   98.2

  2-6
      4*m=240 Trough only     Low    1.37	   5.01	   98

           Medium    1.21	   4.69	   97.9

           High    1.1	   4.48	   98.5

           Perfect    0.77	   3.94	   98.8

  2-6
        4*m=120 Full     Low    0.64	   7.95	   99.3

           Medium    0.43	   7.27	   99

           High    0.81	   6.46	   98.8

           Perfect    0.27	   5.92	   99

  16*β_BW.CL     4*m=240 First half     Low    -2.11	   23.94	   90.3

           Medium    -1.6	   22.91	   89.7

           High    -0.68	   20.43	   91.4

           Perfect    -1.14	   18.12	   93

  2-6
      4*m=240 Mixed half     Low    -4.17	   27	   87.7

           Medium    -2.49	   24.61	   89.6

           High    -1.09	   21.12	   92.2

           Perfect    -0.63	   16.9	   94.3

  2-6
      4*m=240 Trough only     Low    -3.73	   33.35	   85.6

           Medium    -5.87	   33.07	   86

           High    -3.31	   30.86	   88.3

           Perfect    -2.91	   25.49	   89.8

  2-6
        4*m=120 Full     Low    -9.1	   35.43	   86.17

           Medium    -7.43	   32.67	   87.2

           High    -4.72	   28.63	   90.4

           Perfect    -5.11	   25.98	   92.2

  16*β_BW.V_c     4*m=240 First     Low    -2	   54.42	   96.7

             Medium    -1.1	   54.68	   96.2

           High    0.84	   50.76	   95.6

           Perfect    2.45	   45.73	   95.2

  2-6
      4*m=240 Mixed half     Low    -5.88	   58.98	   96.7

           Medium    -2.13	   55.02	   95.8

           High    -1.15	   52.32	   95.7

           Perfect    2.98	   45.23	   94.3

  2-6
      4*m=240 Trough only     Low    2.14	   68.17	   94.8

           Medium    -2.06	   63.8	   95.1

           High    2.49	   64.63	   94.4

           Perfect    1.45	   56.91	   96.3

  2-6
        4*m=120 Full     Low    0.83	   70.7	   98

           Medium    -0.23	   66.91	   97.3

           High    5.68	   64.54	   96.5

           Perfect    1.52	   58.06	   97.4








Figure S1: Relative bias of each random-effect PK parameter estimate under complete schedule designs with different sample sizes (m = 60, 120, and 240). (b_CL), (b_Q), and (b_V_p) are the variances of the random effects for CL, Q, and V_p, respectively. (b_CL, b_Q), (b_CL, b_V_p), and (b_CL, b_V_c) are the covariances between the respective random effects. σ^2_1 and σ^2_2 are the proportional and additive error variances, respectively.




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Figure S2: Relative root mean squared errors of each random-effect PK parameter estimate under complete schedule designs with different sample sizes (m = 60, 120, and 240). (b_CL), (b_Q), and (b_V_p) are the variances of the random effects for CL, Q, and V_p, respectively. (b_CL, b_Q), (b_CL, b_V_p), and (b_CL, b_V_c) are the covariances between the respective random effects. σ^2_1 and σ^2_2 are the proportional and additive error variances, respectively.




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Figure S3: Shrinkage estimates under complete schedule design scenarios. b_CL, b_Q, and b_V_p are the random effects for CL, Q, and V_p, respectively.




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Figure S4: Relative bias of each random-effect PK parameter estimate under coarsened schedule designs with m = 240 compared to the complete schedule design with m = 120. (b_CL), (b_Q), and (b_V_p) are the variances of the random effects for CL, Q, and V_p, respectively. (b_CL, b_Q), (b_CL, b_V_p), and (b_CL, b_V_c) are the covariances between the respective random effects. σ^2_1 and σ^2_2 are the proportional and additive error variances, respectively.




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Figure S5: Relative root mean squared errors of each random-effect PK parameter estimate under coarsened schedule designs with m = 240 compared to the complete schedule design with m = 120. (b_CL), (b_Q), and (b_V_p) are the variances of the random effects for CL, Q, and V_p, respectively. (b_CL, b_Q), (b_CL, b_V_p), and (b_CL, b_V_c) are the covariances between the respective random effects. σ^2_1 and σ^2_2 are the proportional and additive error variances, respectively.




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Figure S6: Shrinkage estimates under coarsened schedule designs with m = 240 compared to the complete schedule design with m = 120. The `First half' design samples the first 11 time-points (excluding time 0) out of the total 22 complete schedule time-points. The `Mixed half' design samples time-points after every other infusion. The `Trough only' design samples only trough time-points. All 3 coarsened schedule designs always include the 5-day post 2^nd infusion time-point. b_CL, b_Q, and b_V_p are the random effects for CL, Q, and V_p, respectively.




    < g r a p h i c s >






[pages=1-]Visit_Scheduling_and_Coding_and_Appendix


