Background
The Global Technical Strategy for Malaria 2016–2030 aims to reduce the incidence of new malaria cases by at least 90% by 2030 [
1]. Among the tools that could assist achieving this goal are vaccines that prevent malaria parasite growth in the blood, that is, when the parasite is in the blood stage of its lifecycle. A reliable method to assess the efficacy of blood-stage vaccines is fundamental to decide which candidates are worth further development. A standard methodology to evaluate the activity of blood-stage vaccines is measuring the parasite growth rate, from which the parasite multiplication rate (PMR) can be derived [
2]. Parasite growth rate can be estimated in controlled human malaria infection studies [
3,
4].
The induced blood stage malaria (IBSM) model is a type of controlled human malaria infection in which subjects are inoculated with blood-stage parasites. The inoculum size can be controlled and therefore all subjects in the study can be inoculated safely and uniformly [
5]. Parasitaemia in the blood of subjects is monitored by quantitative PCR (qPCR) [
6], which allows timely data collection from study subjects to estimate parasite growth rate at low levels of quantitation. Efficacy of blood-stage vaccines can be assessed in IBSM studies by determining the reduction in parasite growth rate in the treatment group compared to the control group. As a result, the IBSM model is been increasingly used to test the efficacy of blood-stage vaccine candidates [
7,
8].
Statistical approaches to estimate parasite growth rate include either log-linear or sine-wave models fitted to the log
10 parasite counts over time [
4,
9]. The models used in published IBSM studies [
7,
8,
10] have fixed the intercept on the y-axis to the inoculum size administered to subjects in a given cohort as determined by qPCR. Given that the first parasite counts are only detected by qPCR around 4 days after inoculation, the intercept is fixed to the inoculum size by extrapolating the parasitaemia curve to day 0, which is outside of the range of available data, that is, from day 4 to day 7 or 8, the day when the first anti-malarial treatment is typically given. This extrapolation presumes that parasites would grow log-linearly from day 0 to day 4 and at the same rate as in the measured growth period. Extrapolating the available data from day 4 to day 0 generates a highly influential point (a point of high leverage) assumed to be measured without error, which could bias estimation of the parasite growth rate.
Additionally, models with the intercept fixed to the inoculum size assume that the starting circulating parasitaemia equals the inoculum size and is known for each individual. Although the preparation of the inoculum can be standardised and its size quantified, the actual number of viable parasites introduced into the blood stream of each subject cannot be known with certainty and may be influenced by a number of factors. For example, the time interval between thawing of parasite vials and injection into subjects varies, both within a cohort and between cohorts. Thus, the loss of parasite viability over time would result in some variation in the inoculum size administered to each subject. Moreover, variations in the process of inoculum preparation may result in differences in the inoculum size between cohorts. Hence inoculum size is a controlled variable rather than a constant.
An accurate estimation of the parasite growth rate is paramount to assess the efficacy of vaccine candidates against malaria. In this report, data from three published IBSM studies in which the parasite growth rates were estimated using models with the intercept fixed to the inoculum size was re-analysed [
7,
8,
10]. A set of statistical models was fitted to the published data, including both a fixed and a non-fixed intercept approach, and the estimated parasite growth rates compared.
Discussion
In the present study, 12 different statistical models were fitted to data from three previously published studies to identify the optimal model for estimation of the parasite growth rate in IBSM studies. The analyses show that fitting log-linear and sine-wave models to data without fixing the intercept to the inoculum size results in smaller variability of the parasite growth rate estimates between studies than fitting models with the intercept fixed. This decrease in variability was observed in models fitted by individual and overall by study. The results of this study suggest that the parasite growth rate is similar regardless of inoculum size, which is consistent with the understanding of the biology of parasite growth.
The variability of the parasite growth rate estimates for models fitted by individual within a study is lower in fixed than non-fixed intercept models, which may be an artefact due to the high leverage of the fixed intercept. When the intercept is fixed to the inoculum size, the parasite growth rate is forced to be similar for all subjects within a study, hence reducing the variability across individual parasite growth rate estimates. Variability of the parasite growth rate is crucial for calculation of sample size of IBSM studies: the lower the variability, the smaller the required sample size. Therefore, it is important that variability of the parasite growth rate is correctly estimated and generalizable to the larger population, that is, not study specific.
The error associated with the parasite growth rate estimates for models fitted overall by study using mixed effects models appears also lower for fixed than for non-fixed intercept models. However, as detailed in Marquardt et al. [
14], comparing the error from fixed and non-fixed intercept models estimated using mixed effects is not appropriate.
The results presented in this report confirm previously reported findings that parasite growth rate estimates are similar in log-linear and sine-wave models [
15] and that log-linear models are functionally equivalent to sine-wave models when evaluating parasite growth rate. Sine-wave models provide useful additional information on the periodicity and amplitude of the in vivo growth of
P. falciparum. For computational purposes, fixing the intercept to the inoculum size can facilitate modelling as estimating one fewer parameter can reduce difficulties with model convergence. This is particularly true for sine-wave modelling, where even after fixing the period to 48 h the model will still require five or more data points per subject to estimate all parameters. Nevertheless, sine-wave, non-linear mixed effects models allow all data points to be included, even if fewer than five data points are available for some subjects.
Modelling the data by individual or overall by study had minimal effect on parasite growth rate estimates in malaria-naïve subjects. Mixed effect models used to fit data overall by study combine data with appropriate weights across individuals to estimate an overall parasite growth rate. In models fitted by individual, averages of the individual fits are an unweighted version of the same analysis. Hence, it is not surprising that the analyses performed either by individual or overall by study give very similar estimates of the in vivo growth rate of the P. falciparum 3D7 parasite. Based on simplicity and greater flexibility, the individual fits are preferred over overall study fits. Moreover, individual fits allow investigation of individual immune factors, which are of interest in vaccine trials. However, if the subjects differ greatly in number of data points or have very few points available for modelling because of logistical issues, a weighted average of the individual fits should be considered.
A number of biological reasons further support the rationale for not fixing the intercept when estimating the parasite growth rate. Fixed intercept models assume that the number of viable parasites in the inoculum is constant, both between study subjects in an individual cohort and across studies. However, there is a paucity of experimental data to support this hypothesis. Moreover, a range of sources indicate that parasites may grow at different rates in different subjects, depending on factors such as the subject age, immunological response and red cell factors that may influence parasite replication [
16‐
18]. Thus, extrapolating data from day 4, when parasites are initially detected by qPCR, to day 0, may introduce a confounding effect that is numerically substantial and lacking in biologic plausibility. By not fixing the intercept, this potential confounding effect is accounted for. Therefore, parasitaemia at day 0 for individual study subjects is more accurately estimated using non-fixed intercept models.
The analyses presented in this report were slightly different compared with the original reports. Whether the intercept was fixed to the same inoculum size as in the original reports is not certain. However, since this study closely reproduces the parasite growth rate estimates reported in each of the original publications, the differences between the analyses are not critical for the conclusions of this study.
Authors’ contributions
LW data analysis and interpretation, and writing the draft manuscript. IH data management and analysis. LM guidance of mixed model fitting algorithms and interpretation of model parameters. PO study formulation and interpretation of model parameters. JSM interpretation of model parameters. All authors read and approved the final manuscript.