New Mathematical Models of Antimalarial Drug Action to Improve Drug Dosing Regimens

  • James M. McCawEmail author
  • Pengxing Cao
  • Sophie Zaloumis
  • Julie A. Simpson
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 28)


Plasmodium falciparum malaria remains a major threat to global public health. Artemisinin-based combination therapies—a critical component of current control strategies—are at risk of failure due to the emergence of artemisinin resistance. To extend the life of artemisinin-based therapies, it is crucial that we develop a better understanding of how they act to reduce parasitemia in the host. Recent laboratory-based experiments have demonstrated that parasites respond to the cumulative, rather than instantaneous, drug concentration. This observation directly challenges the standard paradigm of pharmacokinetic–pharmacodynamic (PK–PD) modelling. Here, we introduce a generalisation to the PK–PD model which accounts for cumulative exposure. Parasites accumulate ‘stress’, which translates into an effective killing rate which can vary with both drug concentration and exposure time. Our model indicates how drug-resistant parasites may avoid killing. Through simulation, we explore alternative drug dosing strategies that may overcome drug resistance.


Mathematics for Industry Biological modelling Malaria Antimalarial drugs 

1 Introduction

Artemisinin derivatives (ART) provide the first-line treatment for falciparum malaria, a major parasitic disease affecting millions of people every year [1]. Their extensive use over the past decade has dramatically reduced the burden of malaria on human populations. However, over recent years, clinical signs of drug resistance have become established in South-East Asia and ART therapy is now at risk of failure [2].

Pharmacokinetic–pharmacodynamic (PK–PD) models—which combine models of blood antimalarial drug concentrations with models of parasite replication dynamics—have been used extensively to study the mechanisms of action of drugs, interpret clinical trial data on alternative dosing regimens and guide the development of drug dosing policy [3].

Laboratory experiments, conducted by collaborators at the Bio21 Institute (Melbourne), have established that the dynamics of drug killing are complex [4, 5]. Not only do parasites display stage-specific sensitivity to ART, the rate of parasite killing appears to depend upon the cumulative exposure of parasites to drug, rather than the instantaneous drug concentration. Furthermore, in experiments with ‘drug-resistant’ parasite strains, a clear loss of sensitivity to the drug was observed during the ring stage of the parasite’s life cycle.

In this context, understanding the mechanisms and dynamics of drug-induced parasite killing requires the development of new PK–PD models of drug activity and parasite response [3]. The development of improved models may prove crucial in optimising drug regimens to either overcome or delay the onset of drug resistance and improve clinical outcomes.

Here, we introduce a model of parasite killing in the presence of a time-varying drug concentration and extend the PK–PD modelling paradigm to account for drug accumulation effects.

2 Model

During blood-stage Plasmodium falciparum infection, the parasites go through a 48-h asexual reproductive life cycle. We consider the number of parasitised red blood cells, N(at), of age a at time t to evolve according to
$$\begin{aligned} \frac{\partial N(a,t)}{\partial t} + \frac{\partial N(a,t)}{\partial a} = -kN(a,t) \;, \end{aligned}$$
where k is the (drug-induced) parasite killing rate, which in general will be a function of the detailed history of drug exposure. The domain of a is the time from formation of an infected red blood cell to its time of rupture \(a_r\) (usually 48 h). We have a boundary condition for asexual reproduction \(N(0,t) = rN(a_r,t)\) (\(r \approx 10\)), indicating that parasites released from a single ruptured red blood cell infect (on average) 10 susceptible red blood cells.
From this general formulation (which will be used to perform simulations of in vivo parasite dynamics at the end of this paper), we simplify to consider a tightly age-synchronised population of parasites of age \(\bar{a}\) as was used in the in vitro experiments of Klonis [4] and Dogovski [5]. Drug responses are (empirically) observed to be well described by Hill function kinetics:
$$ k(C) = \frac{k_\text {max} C^\gamma }{K_c^\gamma + C^\gamma } \;. $$
To capture the effects of cumulative drug exposure, we model the maximal killing rate, \(k_\text {max}\) and half-maximal concentration \(K_c\) to be functions of an accumulated parasite ‘stress’, S, which increases as follows whenever drug, C(t), is present at a concentration higher than some (small) critical value, \(C^*\):
$$ \frac{dS}{dt} = \lambda (1-S) \;. $$
The stress, S, is immediately reset to zero when C(t) drops back below \(C^*\).
With \(k_\text {max} = \alpha S\) and \(K_c = \beta _1 (1-S) + \beta _2\), with \(\alpha \), \(\beta _1\) and \(\beta _2\) positive constants, and assuming the presence of drug at a concentration \(C(t)>C^*\) for the entire experimental assay, we obtain
$$\begin{aligned} k_\text {max}&= \alpha \left( 1- e^{-\lambda t} \right) \\ K_c&= \beta _1 e^{-\lambda t} + \beta _2 \;, \end{aligned}$$
and so the number of parasites \(\bar{N}(t)\) (of initial age \(\bar{a}\)) surviving at time t is given by:
$$ \bar{N}(t) = \bar{N}_0 \exp \left[ -\int _0^t k(C(\tau ),S(\tau )) \; d\tau \right] \;. $$
The in vitro experiments [4, 5] exposed parasites to drug pulses of a particular duration (\(T_d\)) and particular (initial) concentration \(C_0\). The half-life of in vitro drug decay was also measured. Rather than measure counts of parasites directly, the experiments provide a relative measure of parasite survival based on the number that survive until rupture, producing ‘offspring’ (with expansion factor r (see above)) in the next generation. This measure is called the viability and is constrained to lie in [0, 1]. After some manipulation (and with details of the experimental procedure [4]), it can be shown that the viability is given by
$$\begin{aligned} V(C_0,T_d) = \exp \left[ -\int _0^{T_d} k(C(\tau ),S(\tau )) \; d\tau \right] \;. \end{aligned}$$

2.1 Results

We used model (2) to fit the available age/stage-specific in vitro data. For each of the four parasite stages (early rings, mid-rings, early trophozoites and late trophozoites), we obtained (stage-specific) estimates for \(\lambda \), \(\alpha \), \(\beta _1\) and \(\beta _2\). With these estimates—which show strong evidence for drug accumulation effects (i.e. \(\lambda \) is small, data not shown)—we then simulated realistic in vivo PK–PD curves and explore the effects of drug accumulation.

Figure 1a shows how the overall killing rate k evolves for the mid-ring stage of the parasite life cycle given a typical PK (drug concentration) time profile. The trajectory in black is for \(\lambda = 0.37\), as estimated from the in vitro data. Curves for slower accumulation of ‘stress’ (\(\lambda = 0.1\), red) and more rapid accumulation (\(\lambda = 1.0\), blue) are also presented to highlight the potential biological importance of drug accumulation. It has been suggested that drug resistance may manifest as an increased tolerance to drug for the parasite [5]. We model this possibility as a reduction in \(\lambda \) for the mid-ring stage of the life cycle. Figure 1b illustrates the dramatic effect on the parasite load over time resulting from this increased drug tolerance for a realistic scenario of multiple drug doses (following the standard drug regimen as recommended by the World Health Organisation).
Fig. 1

a The killing rate surface as a function of drug concentration C and accumulated stress S and a projection of the trajectory for the effective killing rate on that surface for three values of the stress accumulation rate \(\lambda \) (details in text). b Simulation of the parasite load (from simulation of the full model (1)) under a standard 24 hourly treatment regimen for two strains: drug-sensitive 3D7 (\(\lambda = 0.37\)) and a hypothetical drug-resistant strain (\(\lambda = 0.1\))

Having established how an increased tolerance to stress manifests as a delay in clearance, we now use the model to explore alternative drug dosing regimens. Figure 2 presents results for two widely suggested alternative regimens: increasing dose concentration and increasing dose frequency. We apply these alternative dosing regimens to a simulated 3D7 infection (i.e. \(\lambda = 0.37\)). For an initial parasite distribution in the host that is primarily rings (a) or trophozoites (b), we observe a clear benefit in twice daily dosing. In contrast, marginal benefit is obtained through increased dose concentration (4mg/kg vs. 2mg/kg).
Fig. 2

A comparison of alternative drug regimens. a Parasite counts over time for the baseline (2mg/kg per 24 h; red), increased dose concentration (4mg/kg; black) and increased dose frequency (2mg/kg per 12 h; green). Increasing the dose has a minimal effect on parasite count as the drug concentration is sufficiently high in the baseline scenario. However, increasing the dosing frequency to twice daily has a dramatic and positive impact, shortening the time to resolution of infection by between 12 and 24 h. b As in (a) but for an older initial parasite distribution (primarily trophozoites). The improvement obtained by twice daily dosing is even more dramatic

2.1.1 Conclusions

Based on detailed in vitro experiments and an extension to the traditional PK–PD modelling framework, we have explored the potential role for drug accumulation effects in antimalarial activity. Our findings provide new insight into the mechanisms of drug-induced parasite killing and an enhanced predictive platform for evaluating the likely efficacy of alternative ART dosing regimens.



We thank Leann Tilley and her team (Bio21, The University of Melbourne) for access to data. Pengxing Cao and Sophie Zaloumis were supported by National Health and Medical Research Council project and Centre for Research Excellence funding.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • James M. McCaw
    • 1
    Email author
  • Pengxing Cao
    • 1
  • Sophie Zaloumis
    • 2
  • Julie A. Simpson
    • 2
  1. 1.School of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia
  2. 2.Melbourne School of Population and Global HealthThe University of MelbourneMelbourneAustralia

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