The PK/PD modelling now allows for artemisinin absorption and conversion (described above), so the ability to track more than two drug concentrations simultaneously and convert them into a drug-killing rate is crucial. This feature is absent from previous pharmacological models of malaria, which track only a single drug [1] (
link) although we previously extended the methodology to track up to two drugs [13] . Existing pharmacological models typically use a standard differential equation [1] (
link) to find a mathematical description for the rate of change in total parasite growth and death rates where
P is the number of parasites in the infection,
t is time after treatment (days),
a is the parasite growth rate (per day),
f(C) represents the drug-dependent rate of parasite killing which depends on the drug concentration
C, and
f(I) the killing resulting from the hosts background immunity.
As antimalarial drugs are now typically deployed as combination therapies and as each drug may affect parasites in its unconverted and/or converted forms, predicting the changing numbers of parasites requires an expansion of
Equation 9 where
r is the number of drugs, the drug effect
f(Cd) is the effect of each drug,
d. Note that we regard each active entity as a distinct “drug”. For example artemether-lumefantrine (AR-LF) includes three drug forms lumefantrine (LF), artemether (AR) (unconverted) and its active metabolite DHA (dihydroartemisinin). Note that
Equation 10 assumes drugs kill independently; this is discussed further below.
Integrating
Equation 10 allows us to predict the number of parasites at any time,
t, after treatment with any number of drugs. This was done by first integrating
Equation 9 using the separation-of-variables technique
Integrating both sides of
Equation 11 gives so
Taking the exponential of both sides (and noting that
a times
0 = 0) gives so
The problem is now to integrate
f(C). Assuming there are
r separate drugs/metabolites with antimalarial activity. In this case,
f(C) becomes
So for each drug/metabolite
d we need to calculate its concentration over time
Cd using the compartment model
Equations (7 and
8) and the substitute
Cd into the killing rate equation
Note in
Equation 14, is the maximum drug killing
Vmax for drug
d.
Substituting
Equation 13 into 12 gives or, equivalently,
Note that
Cd may be a complicated expression (including
Equations 7 and
8) and so has to be integrated numerically. As before [13] , if the predicted parasite number (
Pt) falls below 1 we assume the infection has been cleared and the patient cured, immunity is currently ignored (see Winter & Hastings [13] for further
discussion).
Kay K, & Hastings I.M. (2013). Improving Pharmacokinetic-Pharmacodynamic Modeling to Investigate Anti-Infective Chemotherapy with Application to the Current Generation of Antimalarial Drugs. PLoS Computational Biology, 9(7), e1003151.