Investigating the Efficacy of Triple Artemisinin-Based Combination Therapies for Treating Plasmodium falciparum Malaria Patients Using Mathematical Modeling

The first line treatment for uncomplicated falciparum malaria is artemisinin-based combination therapy (ACT), which consists of an artemisinin derivative coadministered with a longer-acting partner drug. However, the spread of Plasmodium falciparum resistant to both artemisinin and its partner drugs poses a major global threat to malaria control activities.

2 Modelling combined killing effect

Models of drug interaction
There are two prominent empirical approaches for modelling zero-interaction: Loewe additivity (1) and Bliss independence (2). Loewe additivity is based on the idea that two non-interacting drugs differ only in their potency, and was originally formulated as where c 1 and c 2 are the concentrations of drugs 1 and 2, respectively, that each individually (i.e. not in combination) produces a specified effect E 12 , and C 1 and C 2 are the drug concentrations in a combination that together produce E 12 -for brevity, the formulae are defined for two drugs, but they can be readily extended for multiple drugs. Eqn.
(2.1) is known as a linear isobole, which is widely used in pharmacology and toxicology as a reference to identify drug interactions. Loewe first put forward this model, which was then investigated more rigorously by Berenbaum (1985) and others.
Loewe additivity is suggested to be a suitable concept for zero-interaction when the combined drugs have similar modes of action (4,5). However, when the drugs are believed to act independently, Bliss independence is more appropriate. This model is based on a probabilistic perspective, defined as where E 1 and E 2 are the individually produced effects by drugs 1 and 2, respectively.
Ultimately, deviations from a selected zero-interaction reference model would determine the degree of synergistic/antagonistic interaction in certain drug combinations. Note that despite the fundamental differences of Loewe additivity and Bliss independence, it has been shown that they indicate the same nature of drug interactions in the majority of cases (6).

Combined effect of DHA-PPQ-MQ
Statistical models can be used to define E P M , e.g.
where C P and C M are the concentrations of PPQ and MQ, respectively, and β 0 , . . . , β 3 are the coefficients of the model. Similar statistical models can be found in (8,9).
Another set of models include only one parameter to incorporate the effect of interaction (4, 10, 5). These models are more specified to the framework of drug interaction, in contrast to the statistical models. Here, we focus on the models with one parameter of interaction -noting that statistical models are shown to be readily transformable to these models, e.g. see (7).
One of the most frequently used models to describe the combined effect is Greco's model (4), defined by where the subscripts P and M denote which drug the parameters correspond to. The Then, we find the concentration of PPQ that is equally effective as MQ at concentration C M , using where E −1 P is the inverse function of E P , given by but, the resultant E P M is non-monotonic, which is biologically infeasible. We also tried other terms such as α C P C eq,M , but they similarly failed to give either a good fit or a monotonic effect. Hence, the models of form Eqn. (2.3) did not produce an appropriate E P M , as also outlined by White et al. (2003) and Machado, Robinson (1994).

Then, the zero-interaction model is obtained via
We then turned to using the model introduced by Machado, Robinson (1994): data (see Fig. 5a), and importantly, a biologically feasible killing effect, E P M (see Fig.   5b). Therefore, we selected this model for E P M , and used it in the combined effect, Eqn.
To conform with the data provided by Davis et al. (2006) (13), the maximum killing effects and sigmoidicity of PPQ and MQ are considered equal (i.e. E max,P = E max,M = 0.3 and γ P = γ M = 3) throughout the model fitting. However, the considered range of variation for α in the simulations is significantly larger than the potential variations due to E max,P = E max,M and/or γ P = γ M , hence, these assumptions do not invalidate the results (see Table 3).
3 Calculating E max using the parasite reduction ratio (PRR) We are interested in finding how E max is related to the parasite reduction ratio (PRR).
We can estimate PRR by where T is the time when we count the number of parasites (e.g. T = 48 hrs) to calculate PRR, and N 0 is the initial number of parasites at time t 0 . Then, we have where a τ = [(a + τ ) mod 48]. Thus, we use numerical methods to solve the above equation for E max . The estimated E max values are listed in Table 3. Note that it is extremely important to take account of the details of the clinical efficacy studies, by which the PRRs of the drugs are obtained. We used the following PRRs and the dosing regimens to estimate E max for each drug: • PRR DHA = 10 4 : seven 2 mg/kg doses of DHA are administered (14).
The obtained E max is then used as the median of the triangular distribution (see Table   3). The lower (E max,l ) and higher (E max,h ) limits of the distribution are assumed that correspond to 50-fold increase and decrease in the above PRRs, respectively, which yields E max,l = E max − log(50) ||W || , where ||W || is the size of killing window of the drug (16).