Background
The complex life cycle of
Plasmodium falciparum involves many parasite stages within both the mosquito and human host.
Plasmodium falciparum induces complex and non-sterilizing immune responses with repeated possible exposure in both the human host and mosquito. The human blood-stage infection plays a crucial role in both disease burden and transmission. Indeed, the length and magnitude of asexual parasite infection both drive clinical symptoms within a host and the transmission potential through the level of gametocytes. Thus, understanding within-host dynamics of the asexual parasite stage is essential for both the development of drugs or other tools that target asexual or gametocytes stages, and to assess burden or transmission dynamics. Within a human host, the
P. falciparum malaria cycle begins with sporozoites transmitted by infectious mosquito bites that travel from the skin to the liver [
1]. Following replication in the liver, merozoites are released into the bloodstream [
1,
2]. These merozoites subsequently infect red-blood cells (RBCs) and in vitro
, approximately 16 merozoites emerge from a single merozoite in a 48 h asexual blood-stage cycle [
3]. The blood-stage infection can persist for over 300 days if untreated [
4]. A small fraction of asexual parasites convert into gametocytes [
2], responsible for the transmission from human to mosquito.
The asexual malaria life cycle drives clinical disease in an individual, with many processes eliciting or evading the immune response [
1]. After invasion of a RBC, merozoites are no longer directly exposed to immune actors. However, with hundreds of parasite proteins exported to the erythrocyte’s cell surface, immune effectors can recognize an infected RBC (iRBC) [
1]. Naturally acquired immune responses recognize erythrocyte surface antigens of iRBCs or antigens of the free merozoites [
1], as well as antigens from liver and sexual stages [
5,
6]. Exported parasite proteins on the cell’s surface give the cell the capability to adhere to the blood vessel’s wall, and thus evade splenic clearance [
1]. Furthermore, expression of the most characterized exported protein, the erythrocyte membrane protein 1 (PfEMP1), can be switched by the parasites from a large library of variants [
1]. New protein conformations are produced to avoid detection, requiring the host to mount new immune responses [
1].
Plasmodium falciparum escapes the immune response by successively expressing one out of 50–60 different PfEMP1 genes [
7]. The switching mechanisms remain uncertain, but switching between the PfEMP1 variants needs to be quick enough to evade the immune system and avoid splenic clearance, while slow enough to avoid variant exhaustion and maintain the chronic nature of the infection [
7]. In endemic areas where populations are continuously exposed to malaria, repeated infections lead to acquired immunity, preventing severe cases of malaria and death but without leading to sterilizing protection for infection [
1].
Many mathematical and statistical models have been developed to understand population level dynamics of malaria transmission and impact of interventions (reviewed in [
8,
9]), or to understand within host dynamics. Although there is a long history of mathematical modeling of malaria parasite within-host dynamics over the years, the substantial biological unknown elements of both parasite and host dynamics and the highly variable nature of infection patterns, make it difficult to assess model accuracy. Furthermore, as there is limited within-host data available for infections from either immunologically naïve or non-naïve individuals, there is no “gold standard” data set or model to compare. In 1999, Molineaux and Dietz reviewed published intra-host models [
10], indicating the first within-host model of malaria was likely developed in 1989 by Anderson et al
. [
11]. Anderson and colleagues [
11] described parasite dynamics via a set of differential equations representing uninfected RBCs, iRBCs, merozoites, and immune effectors [
11]. This model along with the others reviewed led Molineaux and Dietz to conclude that existing models lacked realism and did not make quantitative comparisons to real data. They further concluded that the reviewed models did not allow for inter-individual variability in the outputs, even though a large variation in infection dynamics exists between individuals [
10]. In part to address these concerns, a substantial number of mechanistic within-host models, either standalone or used in larger transmission models, have since been developed. Most of these models were initially parameterized to data from naïve patients, but not necessarily to the limited available data from previously exposed individuals.
Several sources of detailed observations of parasite dynamics and densities in naïve patients exist. In the past, malaria infection was induced to generate fever to treat other illnesses. In particular between 1917 and 1963 malaria was used as a therapy to treat patients with tertiary neurosyphilis before the use of penicillin [
12]. The most extensive malaria therapy data set was collated between 1940 and 1963 by Collins and Jeffery [
12‐
15]. The published database consists of 318 patients treated at Columbia, South Carolina and the Milledgeville, Georgia laboratories [
12]. This data, referred to here as the malariatherapy dataset, includes patients infected with three different strains of
P. falciparum for neurosyphilis treatment. The data captures daily parasite counts by microscopy of both gametocyte and asexual parasites, and daily fever charts are available for each patient. This data set is the only detailed representation of the entire
P. falciparum infection in a naïve population. Other malariatherapy data sets exist, for example of
Plasmodium vivax infection in naïve and non-naïve individuals [
16] (not further discussed here).
Over the last decade, many individual-based models of malaria transmission dynamics have been developed (reviewed in [
8]); several of these include models of within-host asexual parasite dynamics [
17‐
20]. A recently published paper [
21] investigated common biological assumptions made by within-host models, and concluded that current knowledge is insufficient to capture infection lengths and to explain the chronic nature of malaria infections. They further concluded their model was quite sensitive to small changes in the parameters leading to large instabilities in estimated infection lengths [
21]. Since asexual parasite dynamics are particularly important for modelling the effect of malaria interventions targeting humans (such as drugs or vaccines), the within-host model assumptions in these models have the potential to drive predictions at the population level, on either disease burden or intervention impact. Thus, with these within host models widely used in public health research, a good understanding of the overall dynamics, the assumptions, the uncertainties, and limitations of the models are key to critically assess and understand the predictions arising from the use of those models.
In this review, within-host models of asexual parasitaemia were analysed and components of the models which drive predicted dynamics were identified. The identified models were used to investigate malaria interventions such as drugs or vaccines, either as stand-alone within-host models or used in combination with transmission models. The review and analytical assessment of each model, including the re-simulation of a subset of models to allow for deeper investigation, provides an overview of the main components of each model and their underlying assumptions. Rather than defining a gold standard, models were discussed on how they differ in their immune responses and parasite growth. This comparison provides an understanding of the benefits and limitations in using these models, which directs and informs future work on within-host models of blood-stage parasitaemia.
Discussion
In this study, eight published mechanistic within-host models of the asexual blood-stage dynamics of P. falciparum were reviewed, five of which were reproduced via simulation analysis. Several features and simulation outputs from the models were compared including the predicted time-series of asexual parasitaemia, modelled growth rates, innate immune responses, variant-specific immune responses, and general adaptive immune responses. The models varied widely in complexity. Rather simple models such as McKenzie and Bossert have the advantage that they do not rely heavily on assumptions of unknown biological mechanisms, while more complex models, such as Eckhoff or Childs and Buckee capture more detailed, yet less well understood, immune and parasite mechanisms. Understanding the variation in multiplication rates, versus immune and other host factors, or random effects and measurement error, and their impact on parasite density variations is particularly important when the models are included in broader investigations of the effect of a vaccine, drug or other interventions aimed to modify parasite growth patterns. The overview presented here provides a general understanding of those models.
Model composition varies in complexity and uncertainty
Parasite and host dynamics are represented in the mechanistic models via detailed description of the parasite replication dynamics, and up to four host immune responses. Each model has its own additional complexity, specifications, and advantages. For example, to obtain increased detail of the host’s response, some models include red blood cell availability and limit the maximum immune response capacity. Or, to include more details in the parasite dynamics, some models include specific variant switching mechanisms. Models generally define a negative feedback loop between the parasite density or cumulative parasite density since the start of the infection and the effect of the immune responses, and in addition some models add the time of infection (Gatton and Cheng, Childs and Buckee, Eckhoff) as a determinant for the magnitude of the general immune response.
Including stochasticity in the models is particularly relevant for the within host dynamics of P. falciparum as the infection patterns observed in the malariatherapy data are highly variable among patients. Yet this is a challenge for modellers as it requires more complex models that include stochasticity, and it requires capturing inter-individual variation in the fitting process. Molineaux et al. proposed nine summary statistics for the 35 malariatherapy patients as outcomes of interest to be reproduced. Other models have used those summary statistics to describe the dataset, although most did not include all nine. Given the limitations of the malariatherapy dataset, the models should not aim to reproduce all nine features, nevertheless characteristics such as a wide range of infection lengths, a high early peak representing the acute infection phase, sometimes followed by chronic infection, should be accounted for. In the context of within host models used for modelling transmission and impact of interventions, additional key features to capture, which were not included in the current analysis, are infectiousness and symptomatology.
Our analysis highlighted that the Molineaux et al
. and the Molineaux-adapted models have likely allocated too much stochasticity to the individual parasite multiplication rates, thus masking other mechanisms, and placing relatively less importance on immune responses and other host factors. Furthermore, for these models, their assumptions concerning the inherent multiplication rates of the parasites differ from other models, along with assumptions of large variability in the variant-specific parasite multiplication rates in the absence of any immune response. These dynamics were found to be essential in the Molineaux et al
. and Molineaux-adapted models to reproduce clinical malariatherapy patterns of infection, rather than immune responses [
22]. In particular, in Molineaux et al
. and Johnston et al
., longer infections result from the expression of variants with a high multiplication rate towards the end of the infections for later variants. In contrast, the other models did not rely on variation to capture infection patterns. Instead, variation was mainly included in the control of infection due to immune responses and switching mechanisms.
For models including multiple parasite variants, the variant switching dynamics are an important mechanism driving the parasitaemia predictions. The switching dynamics define how the parasite goes from expressing one PfEMP1 variant to another one at the next generation to evade the immune response. Switching dynamics in the models have been assumed to respond to the variant specific immune response (Molineaux et al
. and Johnston et al
.), to the current variant population size (Eckhoff), or were determined by more sophisticated switching networks (Childs and Buckee, Gatton and Cheng). Both the variant switching dynamics and the variant-specific immune response are essential drivers of infection patterns and inter-individual variability in all models (except McKenzie which does not include variants). With detailed
var gene transcription analysis studies limited to early days of infection in volunteer infection studies (VIS) [
33‐
35], data available to inform the models on switching dynamics over the entire course of an infection remain insufficient.
The various assumptions around the mechanisms of action of the different immune responses and the interplay between parasite and host highlight the challenge of realistically reproducing the time-series observed in the malariatherapy dataset, especially the inter-individual variability. This challenge is confounded since there is limited knowledge of the biological mechanisms at play.
Parasite multiplication rates might be lower than initially assumed
It is commonly agreed that a single iRBC produces 16 merozoites [
3], of which a portion successfully invade new erythrocytes. Growth rates in vivo are more difficult to measure, and compared to the assumed multiplication factor of 16, includes the host-parasite interactions, which reduce the observed parasite growth. Several independent statistical models previously estimated parasite growth at onset of infections in both malariatherapy or VIS. Estimates from the malariatherapy dataset range between 10 and 18 [
36], or 6 and 24 [
20]. In malaria VIS the growth factor was estimated to be between 12 and 15 [
37], and in the control cohort of vaccines AMA1-based vaccine challenge between 14 and 21 [
38]. More recently, estimated ex vivo multiplication factors for different malaria genotypes were found to be between 2–11 for laboratory strains and new clinical isolates [
39]. One hypothesis explaining the variation among parasite growth relies on the differential capacity of the PfEMP1-variants to evade splenic clearance, with the hypothesis that a subgroup of PfEMP1 expressing parasites might be fast growing (due to increased cytoadherence and thus decreased splenic clearance) [
40]. This mechanism may explain differing growth rates among parasites expressing different variants, and explain apparent higher multiplication rates in naïve individuals if their parasites express the fast growing PfEMP1 subgroup [
40]. Variance in parasite multiplication rates among clones and among infections across individuals is possible due to this variance in successful avoidance of splenic clearance during blood-stage replication, however, it is unclear whether the range should include an overall multiplication factor as large as 32 or 35, as in some models reviewed here (namely in [
21‐
23,
25]). RBC availability might also constrain successful invasion of RBCs and thus affect the effective replication of the parasite. Recent in vitro studies highlighted distinct RBC invasion strategies of
P. falciparum strains, with parasites that favour RBCs of different age [
41], and different parasite strains either invading a larger fraction of RBCs at lower rates or invading smaller fraction of RBCs at a higher rates [
41]. In addition to potential age-dependent differences in RBC availability, it is known that certain RBC polymorphisms, for example sickle cell traits and blood groups [
42] impact the invasion of RBCs. These studies suggest that variability in effective parasite growth, both within host and across individuals, might be attributable to heterogeneous RBC accessibility and susceptibility to parasite invasion.
Variant switching and immune response modeling are limited by current knowledge
Switching mechanisms are not well understood, and it remains unclear if the switching mechanisms are driven by antibody response (as in [
22]) or are not directly influenced by the immune pressure (as in [
21,
24]). In contrast to the assumptions made in the models reviewed here, it is likely that parasites express more than one variant, if not all variants, during the first blood stage generation [
34]. As highlighted by Childs and Buckee [
21], this finding challenges the current understanding on the underlying mechanisms leading to chronic infections. Cross-reactivity, suggested as a mechanism necessary for chronic infections [
43] and included explicitly in Gatton and Cheng, Childs and Buckee, and in Eckhoff would not allow for chronic infections if all variants are expressed at infection onset [
44], and models would have difficulties to recover long infection patterns. The lack of understanding about the switching dynamics supports the assumptions in Challenger et al
., as they only model total parasitaemia without modeling switching between variants. The models described by McKenzie and Bossert, and Gurarie et al
., are less complex and do not model variant-specific parasitaemia, offering potential modeling alternatives when detailed mechanisms of immune response are not needed. To our knowledge, there are few biological studies on the kinetics and interplay of the immune responses as defined by the models (innate, variant specific, and/or adaptive immune response). As such it is unclear how much each immune component affects the overall time course of infection. Therefore, it is not surprising that models differ in the relative importance of general or variant-specific immune responses. Moreover, in the absence of a clear biological understanding of the variability in infection time-series, model stochasticity remains an important driver of the modelled immune and parasite dynamics.
Further data for new models
Blood samples can only inform on the level of circulating parasites (and associated measures), thus current tools are blind to parasites while sequestered and hidden in capillaries to avoid splenic clearance [
1]. Consequently, it remains difficult in humans to experimentally assess localized host-parasite interactions for a full understanding of the role of different immune actors and potential resource limitation. Nevertheless, a few data sources on circulating parasites are available.
It is essential to highlight here that the malariatherapy dataset cannot be considered as a typical time course of an infection, as the data come from patients with severe neurosyphilis, who were malaria naïve (although previous infection cannot be completely ruled out as some patient lived in area where malaria was endemic at that time), and did not include children. Furthermore, inoculations were limited to a restricted set of parasite strains, measurements were prone to errors, and data did not include any measurements which could directly inform immune response and RBC dynamics, nor parasite gene expression dynamics. Thus, although the malariatherapy data are the only detailed data available on infection time-course, models are not strictly evaluated by their ability to reproduce malariatherapy-like infections. Data on early infections are available in high detail from VIS. They provide precise quantification of parasitaemia at much lower detection threshold than the malariatherapy data, and variant expression dynamics or other genetic traits relevant for understanding the dynamics in the first few days of blood stage infection can now be informed by such studies. Note that most of the parasite densities measured in VIS fall below the 10 iRBC/μl detection threshold in the malariatherapy records, and untreated infections last ten days at most. Thus, most parasitaemia levels available from VIS would probably correspond to pre-recorded infection times in the malariatherapy records, making a direct comparison between the two datasets difficult. Beyond early infections, models need to rely on longitudinal field data to assess their performance. Longitudinal field data are extremely important to explore the dynamics in realistic settings, with individuals living in endemic areas who are repeatedly exposed to malaria, including children who are most at risk for the disease, and including a range of genetic diversity and complexity of infection. Because immunity builds up with age and exposure [
45], and genetic diversity is a result of immune pressure, longitudinal and cross sectional field studies which include genetic analysis give important insights in malaria infection dynamics.
The current analysis and review focused on infections in naïve individuals and did not include a review of the models for their ability to capture infections in pre-exposed individuals. Although data is lacking, the immune effect of pre-exposure could be added to the models as a second step, for example by adding an overall reduction factor that would lower the magnitude of the parasite density in function of age and/or exposure, similar to an empirical model by Maire et al
. [
46]. The effect of co-infection was not included here and its implementation was a focus of the Childs and Buckee’s model [
21], which hypothesizes that co-infections and super-infections have different effects based on the timing of the second infection, and that the effects of multiple infections seem to be poorly understood, and thus poorly included in models [
21]. Recently many field studies focusing of genetic data and analysis are giving insights in the effect and dynamics of multiple infections on a population scale (for example [
47,
48]), yet empirical data on the time course of complex infections are sparse and insufficient to validate models of co-infections, relying on data from mouse models for detailed infection dynamics [
49,
50].
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