Background
Malaria, caused by infection with parasites of the
genus Plasmodium, is responsible for over half a million deaths per year worldwide, with children being the main victims [
1]. The immense severity of the disease and the long coexistence of humans and
Plasmodium parasites have even led to the emergence and perpetuation of possible protective genetic disorders like thalassaemias and sickle-cell disease, as well as haemoglobin C, haemoglobin E, and G6P dehydrogenase and pyruvate kinase deficiencies [
2‐
7]. In primates,
Plasmodium sporozoites infect, transform and multiply within parenchymal hepatocytes to form merozoites; these developmental life cycle processes are asymptomatic for the host. Once merozoite forms of the parasite are released from the infected hepatocytes, the parasite begins its cyclical blood-stage development. Merozoites invade and multiply within red blood cells (RBCs) every 24, 48 or 72 h, depending on the species, and new merozoite progeny are released to invade other RBCs [
8]. During this cyclical process of invasion and destruction of RBCs, the symptoms and clinical complications associated with malaria emerge. The blood stage of an infection is characterized by repeated rounds of RBC invasion which, if not kept under control by host immune responses or anti-malarial treatment, can lead to exponential growth of the parasite, with a concomitant destruction of the parasitized RBCs. This destruction, however, is not the predominant mechanism of RBC removal that leads to anaemia, and indeed seems to be vastly surpassed by the destruction of uninfected RBCs (uRBCs) [
9‐
11].
Infections with
Plasmodium coatneyi, a simian malaria species that is closely related to
Plasmodium knowlesi [
12,
13], mirror the biology and pathogenesis of
falciparum malaria, with severe forms of pathology including anaemia. To study mechanisms of the onset and recovery of anaemia, Moreno et al. [
14] established procedures to measure the turnover of in vivo biotinylated RBCs in rhesus macaques (
Macaca mulatta). These macaques had been experimentally infected for the first time (
i.e., when malaria naïve) with
P. coatneyi infected RBCs and, then again, while partially immune, 9 months after curative anti-malarial drug treatment. Five infected and then re-challenged (semi-immune) animals were compared to five control rhesus with biotinylated RBCs, but no malaria infection. Microscopy-based counts of infected RBCs and haemoglobin levels were monitored daily, and the numbers of biotinylated RBCs were assessed using flow cytometry. This work demonstrated that malarial infections result in an accelerated turnover of uninfected RBCs, and this was most pronounced in the malaria naïve animals. The precise mechanisms causing malarial anaemia are unknown, but have been suggested to be due to multiple possible factors leading to a reduction in circulating RBCs including physicochemical membrane changes, reduced erythrocyte deformability, accelerated erythrocyte senescence, and immunological reactions that cause their removal [
14,
15].
This paper presents a mathematical model that was developed to study RBC dynamics in circulating blood during malaria infections. The model was parameterized using experimental results from Moreno et al. [
14] and implemented as a discrete recursive structure that was previously identified as best suited for this class of problems [
16]. The choice of this framework was based on the need to account for the aging of the RBCs rather accurately, which is problematic in delayed differential and integro-differential equations that do not track age directly, but instead approximate the time passed since the cell was generated. Ordinary differential equations with age classes do not model the aging accurately since inspection of age-classes reveals a distribution of ages within each class. Partial differential equation models like the Lotka–McKendrick age-structured population model [
17,
18] do accurately model the aging of a population, but are difficult to implement, especially if the model includes variables with and without an age-structure and if the aging process is perturbed by events like a malaria infection. The most effective alternative is a discrete recursive framework with age-classes. In a sense, this structure corresponds to the discretization of a PDE model and hence shares all of its properties, but is more easily implemented and faster to solve, with an accuracy that is readily tuned by stipulating a desired time-step.
The model permits the quantification of the production of newly generated RBCs and of the different processes leading to the removal of RBCs in the absence or presence of a
P. coatneyi infection in malaria naïve or semi-immune rhesus macaques. Additionally, two alternative mechanisms of uninfected RBC removal, namely accelerated erythrocyte senescence and immunologic removal, are modelled and compared on the basis of their respective predictions. The results demonstrate that the destruction of uninfected RBCs was the dominant process underlying malarial anaemia in the
P. coatneyi infections reported by Moreno et al. [
14], and that the direct destruction of infected RBCs by the parasite accounted for only about 4 % of the total RBC loss. Beyond this specific result, the model can be employed as a tool for predicting and exploring disease severity and evaluating host-directed interventions. This capability includes the study of other species of
Plasmodium that cause malaria in primates, each with their unique blood-stage biology and pathogenic characteristics [
8].
Red blood cell removal processes
In healthy humans and non-human primates (NHPs), RBCs are produced by the erythropoietic system in the bone marrow, which is under the control of several cytokines including, in particular, erythropoietin (for a review see [
19]). Removal of RBCs is a task for the phagocytic arm of the immune system in response to injury, senescence, or other processes. Injured RBCs are usually removed in the spleen, as they fail to deform and can no longer pass through the microcirculation of the red pulp [
20].
Senescence-driven removal of RBCs typically occurs due to oxidative stress to haemoglobin, which results from the continuous cycling between normoxia and hypoxia. This cycling between oxidative states eventually triggers the formation of methemoglobin which ultimately denatures into haemichromes [
21]. These haemichromes are able to bind to the cytosolic side of AE1 protein (Band 3) and thus to displace ankyrin, which weakens the AE1 connection to the cytoskeleton and eventually results in clustering of AE1 [
22,
23]. On the external side, AE1 protein, which is usually found in dimers, has a low affinity for binding naturally occurring IgG antibodies (NAbs). However, once bound to haemichromes on the inside, AE1 dimers are able to aggregate and bind to the NAbs with enhanced affinity [
24,
25]. NAbs alone are not efficient at promoting RBC clearance, but they are able to activate the classical pathway of the complement system and thereby induce erythrophagocytosis [
26,
27]. It has also been proposed that AE1 recognition by NAbs may depend on the proteolytic degradation of AE1 [
28].
Discussion
This work introduces a mathematical model that allows for the disentanglement of concurrent processes relating to RBC removal and malarial anaemia, based on published experimental data from a longitudinal malaria infection study with NHPs [
14]. The main results present a clearer picture of the life span of RBCs in rhesus macaques and the quantification and age-characterization of the enormous proportion of uninfected RBCs that are eliminated coincident with the rise in parasitaemia and progression of the disease.
Developing appropriate models for such purposes requires a flexible computational framework capable of accounting for large numbers of cells in their proper age classes. In particular, a requirement here was that cells could normally be kept in various age classes for more or less fixed amounts of time, but that they could also skip age classes or remain longer within the same classes in certain situations. One approach of addressing such a situation could be based on delay differential equations (DDEs). However, it was shown elsewhere that DDEs are not flexible enough for modelling malarial anaemia at the cellular level [
16]. Delays could also be generated with ordinary differential equations (ODEs) with age-classes, but these do not represent the transitions between age classes well. In contrast to these standard approaches, discrete recursive equations with age classes are not only able to generate hard delays but can also represent the age structure correctly, as can be seen by the fact that all cells in a given age class have the same age [
16].
Building upon these advantages, a discrete system was constructed with age classes and a hazard function was added to account for death. This structure allowed an effective representation of the age distribution of RBCs. In the absence of cell death according to the hazard function, the cells would have a rectangular age distribution with constant death rate. Instead, use of a data-driven hazard function allowed cell removal to be modelled in a manner that is very close to reality. The hazard function that fitted the experimental results best and was used here is a power-law function, which captures the fact that cells of all ages can die, but that the number of cells dying at a young age is very small. By the same token, the older a cell becomes the more likely it will die, and the power-law hazard function captures this behaviour well (Fig.
3b).
Two mechanisms were tested to explain the removal of uninfected RBCs during a malarial infection by processes beyond the normal physiological age-dependent and age-independent processes and the direct parasitization of RBCs. These mechanisms underlying the loss of uninfected RBCs, sometimes referred to as the bystander effect, corresponded to two hypotheses: (1) that all cells are equally likely to be removed, independent of age; or (2) that the removal occurs by normal senescence, but that during high parasitaemia levels uninfected RBCs age faster. The age-independent mechanism would seem to be the better choice if bystander RBC removal occurs due to age-independent loss of uninfected RBCs. This removal could potentially result from targeting of the RBCs by the immune system. Specifically, upon rupture of the infected RBCs and release of the new brood of merozoites, the intracellular contents of the RBC and the parasites are released into the circulation and can adhere to uninfected RBCs, thereby potentially making them targets for erythrophagocytosis.
By contrast, an increased senescence mechanism would be more appropriate if the processes leading to the removal of uninfected RBCs were in fact age-related, for example, with an increased rate of senescence of the uninfected RBCs in the face of an infection. For instance, the immune response to the infection could lead to an increased level of oxygen radicals, which in turn could trigger an increased rate of RBC senescence.
The age-independent and the increased senescence mechanisms both fit the experimental data equally well (Fig.
7). However, they very clearly predict a different age distribution of the RBCs, in particular during the peak of parasitaemia (Fig.
7d). While the age-independent mechanism predicts almost no change in the age distribution of RBCs during peak parasitaemia, the increased senescence mechanism predicts a population of RBCs that appears to be much older. It has been reported that uninfected human RBCs co-cultured in vitro with
P. falciparum infected RBCs have a higher proportion of older cells than same-donor control cultures [
38]. Although a similar trend has been observed here (Fig.
7) based on in vivo data, the results [
38] suggest a different magnitude of increase in older cells, which in turn implies that increased senescence is neither likely the only—nor a major—process leading to the increased removal of uninfected RBCs.
Thus, the age-independent mechanism appears to be more likely an explanation for quantifying the removal processes. Considering this mechanism, the model allows the differentiation between four distinct processes of RBC removal. Two of these processes are normal physiological processes of RBC loss occurring in healthy macaques, namely, age-dependent and age-independent ‘random’ removal of RBCs. The age-dependent process collectively captures the normal processes of RBC senescence, whereas the age-independent processes encompass all RBC losses that occur under normal physiological conditions but are not related to age. This process was modelled as proposed by Löffler’s group [
16,
32,
33], where 10 % of all produced RBCs are destined to be lost by this ‘random’ process.
The remaining two distinct processes of RBC removal are directly and indirectly due to the malarial infection. These are the direct parasitization with the consequential destruction of the infected RBCs and the loss of uninfected RBCs as a bystander effect. RBCs removed by direct parasitization are those RBCs that are invaded by a merozoite for the production of the next generation of merozoites, and destroyed concomitantly with their release into the bloodstream. The estimation of the number of infected cells in the current model is based on the experimental parasitaemia measurements by Moreno et al. [
14]. Uninfected RBC removal occurs during the malaria infection for reasons that are not well understood and clearly warrant further investigation, because this process dominates the manifestation of malarial anaemia, as exemplified by the case studied here of
P. coatneyi infecting
M. mulatta. In both malaria-naïve and semi-immune infected macaques, the loss of uninfected RBCs emerged as the leading process in RBC removal (76 and 67 % of total removal, respectively), whereas the actual destruction of RBCs by parasitization represented only 4 and 1 % (respectively) of the total removal. The remaining losses were 15 % to senescence, and 4 % to age-independent processes, in infected naïve macaques.
RBC production during the 30-day infection of the malaria-naïve macaques was increased by a factor of almost two, which however did not match the total number of cells lost during this period (Fig.
4). The cause of this discrepancy is reflected in the experimental data, as some of the macaques did not fully recover from their anaemic state within the 30-day experimental interval and after 30 days still exhibited decreased haemoglobin levels (Fig.
1). A much smaller difference between the removal and replenishment of RBCs was also inferred for the semi-immune macaques (Fig.
4), as these macaques were mostly able to recover from their anaemias (Fig.
1). Comparison of the inferred RBC production profiles with the experimentally determined percentage of circulating reticulocytes (Fig.
5) showed good agreement, which attests to the model’s ability to predict correctly when erythropoiesis or release of reticulocytes from the bone marrow was upregulated. Yet, closer analysis reveals that this upregulation does not kick in during the time period when RBCs are being actively removed, but only once parasitaemia decreases due to host immune responses or anti-malarial treatment. This discrepancy suggests that the parasite does not reduce erythropoietic output, but that it rather prevents effective erythropoiesis with the normal release of reticulocytes in response to anaemia; only after the parasite is cleared from circulation is the host able to properly respond to the anaemia. Clinically, the infected macaques (whether naïve or semi-immune) showed evidence of erythroid hyperplasia (expansion of erythroid progenitors) in their bone marrow, coincident with the observed peaks of parasitaemia. However, the erythroid hyperplasia was neither reflected by a high number of reticulocytes in the peripheral blood (experimentally measured in [
14]) nor by an increased production of RBCs (inferred in this work), corroborating the hypothesis of parasite-induced ineffective erythropoiesis. Consistent with ineffective erythropoiesis, bone marrow biopsies in naïve macaques showed histological signs of dyserythropoiesis in the form of defective development of erythroid cells [
14]. Relevantly, macaques that had been malaria naïve when infected exhibited erythropoietin levels that were elevated six-fold during periods of high parasitaemia, indicating that ineffective erythropoiesis is not mediated by the dysregulation (i.e., lack) of compensatory erythropoietin production [
14].
It appears that this article is the first to quantify the percentage of removal of uninfected RBCs accurately in an in vivo primate malaria system. It is moreover likely that the results have implications for understanding anaemia in humans that is caused by
P. falciparum (and perhaps other species of
Plasmodium). Two earlier, somewhat indirect studies have attempted to quantify the removal of uninfected and parasitized RBCs in humans infected with
P. falciparum [
10,
11]. Jakeman et al. [
10] used a discrete mathematical model to calculate the ratio of uninfected erythrocytes destroyed per infected erythrocyte in twelve neurosyphilis patients who were infected with different strains (McLendon and El Limon) of
P. falciparum as a treatment (malariotherapy). The results showed that on average 8.52 (1.28–19.28) uninfected RBCs are removed per infected RBC [
10]; thus, the loss of uninfected RBCs represents about 90 % of the total removal. Price et al. [
11] compared the total loss of RBCs inferred from the decrease in haematocrit, with an estimation of the number of RBCs infected and destroyed by parasites inferred from the parasitaemia level. The authors concluded that parasitization was responsible for 7.9 % (6.2–9.6 %) of the RBCs lost in patients with uncomplicated falciparum malaria [
11]. These results are in the same range as our average estimates of 95 % (90–97 %) and 99 % (97–99 %) loss of uninfected RBCs in malaria-naïve compared to semi-immune macaques infected with
P. coatneyi, relative to the total loss of RBCs induced by the infection.
The physiological basis for the high loss of uninfected RBCs is not known. In this paper, two mechanisms are compared: one where any uninfected bystander RBC is prone to removal and one where the infection causes an increased rate of aging and thus the senescence-based removal of uninfected RBCs. If only the second mechanism is in effect, then the model predicts a rather old population of RBCs during high parasitaemia (Fig.
7c, d). This prediction is at odds with experimental evidence obtained in RBC in vitro cultures infected with
P. falciparum (D10 strain), which suggested a slight (~10 %) decrease in young uninfected RBCs and a small (~5 %) increase in old uninfected RBCs in comparison with a non-infected control culture of RBCs from the same donor [
38]. By direct inference, increased senescence would not seem to be the only—or even the predominant—mechanism responsible for the loss of uninfected RBCs during a malarial infection.
Among the age-independent processes that could be responsible for this removal of uninfected RBCs, it has been suggested that reduced deformability, increased oxidation of membranes, inflammatory insults, or surface deposition of parasite proteins like PfRSP-2 could be involved [
39]. Also of interest, data obtained in a model of chronic anaemia in semi-immune BALB/c mice infected with the
P. berghei ANKA strain showed that this increased rate of removal of uninfected RBCs can be delayed by the depletion of macrophages, thus suggesting an immunopathological process where CD4
+ T cells may be involved [
40].
The model put forth here does not account for the microvascular sequestration of parasitized RBCs during the trophozoite and schizont stages of development, a phenomenon that was not experimentally quantifiable with the data, which targeted specifically the dynamic assessment of peripheral blood and bone marrow samples [
14]. The daily monitored, circulating ring-stage infected RBCs can be representative of the parasite burden, but this does not reflect the daily parasite load in the host. Including RBC sequestration in the models would slightly change the rate of death by direct parasitization of RBCs and decrease the rate of removal of uninfected RBCs through the bystander effect. However, the overall results would not change much. As an example, consider the case of malaria-naïve macaques where, during a
P. coatneyi infection, 5 % of RBCs died by parasitization and 95 % were removed as uninfected RBCs. Even if an equal number of infected RBCs were sequestered in the tissue microvasculature and in the peripheral blood circulation, the levels of death by parasitization would merely rise to 10 % and reduce the removal of uninfected RBCs to 90 %. This scenario does not change the main conclusions of this study, and the removal of uninfected RBCs would still greatly outnumber the cell losses due to parasite invasion of the RBCs.
Another assumption of the model, which could be construed as unlikely to occur in humans and in macaques, is the occurrence of random destruction of RBCs as a normal physiological process of RBC loss. This process was here assumed to have a magnitude of 10 %. If the model had been developed without this process, the parameter
c in the hazard function (Eq.
1) would be zero, and the parameters
a and
b would easily compensate for the absence of
c. As a consequence, the survival curve shown in Fig.
3b (blue) would not exhibit the slight slope present between the RBC ages of 0 and 60 days, but would be essentially flat in this range. Nevertheless, given that the same number of cells would still have to be removed, as it is evident from the biotinylated RBC turnover data in Fig.
3a, the hazard function would not change in magnitude, but just in shape. Therefore, as far as the model calculations are concerned, the current removal by a random process would be subsumed into the removal by senescence, while the inferred extent of removal due to the parasite (directly and through the bystander effect) would remain the same. Hence, the blue and purple bars in Fig.
4 would be merged into one single senescence process which, however, would not change the main results and/or conclusions of this work which are predicated on the large difference between the extent of RBC removal by direct parasitization and by the bystander effect.