Luca Barberi1, Sam Jones1, Nathalie Gobeau1
1MMV Medicines for Malaria Venture
Objectives: Developing drugs for malaria chemoprevention requires effective Modeling and Simulation (M&S) approaches to simulate the action of drugs over an extended time-course, ranging from one to several months, during which patients are exposed to multiple infections through mosquito bites. Depending on the mode of action of the drug, the model needs to account for different stages of the parasite life cycle, in particular the liver and blood stages. To represent parasite ageing in the liver before invading the blood, our current methodology utilizes transit compartments [1]. This methodology has some limitations, including numerical diffusion, due to the finite number of transit compartments, and computational cost, due to the high number of compartments required to limit numerical diffusion. We present a novel methodology, based on the analytical solution of liver-stage maturation, that overcomes the limitations of transit compartments while offering a more elegant M&S approach to malaria chemoprevention. Methods: We model liver-stage maturation by using a Partial Differential Equation (PDE) [1]. The PDE takes the form of an advection equation, which is commonly used in fluid dynamics to describe processes like fluid flow through a pipe. Whereas in fluid dynamics the two variables of the PDE are time and space, here the variables are time and parasite age. Upon infection, parasites enter “the pipe” (i.e., the liver) at age = 0, and leave it at age = 6 days when they move to the blood. The PDE can be extended into a so-called reaction-advection form, by including terms that describe parasite replication and death during the ageing process. While more complex than the typical Ordinary Differential Equation (ODE) encountered in pharmacometrics, this PDE has an analytical solution, that can be integrated in our simulations. This type of PDE and its analytical solution are not new in pharmacometrics, where they were used, for instance, in the context of intestinal drug absorption [2]. Results: Compared to our previous methodology based on transit compartments, the analytical solution above has several advantages. From a mathematical point of view, a transit compartment model can be formulated to approximate the PDE described above, whereby the age variable is discretized and the PDE maps onto a system of coupled ODEs. The discretization comes at the cost of a numerical diffusion error [1,3]. Numerical diffusion can be reduced by increasing the number of transit compartments, which also increases the computational cost. The analytical solution does not have this limitation. Furthermore, by significantly reducing computational cost, the analytical solution allows us to separately track the ageing dynamics of parasites coming from different mosquito bites. This is necessary to implement what we call a “cure threshold”, such that, if the total parasite count corresponding to a given bite falls below 1, the parasite density corresponding to that bite is set to zero. Conclusions: By leveraging well-known results for PDEs, we can simultaneously increase the accuracy and reduce the computational complexity of our M&S strategy for malaria chemoprevention. The concepts above can be generally applied to model lag times in pharmacometrics. As noted by other colleagues [3], pharmacometricians would significantly benefit from deepening their understanding of PDEs, to develop more accurate, efficient, and ultimately elegant models.
[1] C. Barcelo, S. Jones, P. J. Lowe, J. J. Möhrle, N. Gobeau, and M. H. Cherkaoui-Rbati, Longitudinal PKPD model of liver and blood stage of P. falciparum malaria to inform late development studies of preventive chemotherapy drugs, In Preparation. [2] P. F. Ni, N. F. H. Ho, J. L. Fox, H. Leuenberger, and W. I. Higuchi, Theoretical model studies of intestinal drug absorption V. Non-steady-state fluid flow and absorption, International Journal of Pharmaceutics 5, 33 (1980). [3] M. H. Cherkaoui-Rbati, The Transit Compartment Model: The Truth behind n!, in PAGE Abstracts of the Annual Meeting of the Population Approach Group in Europe, Vol. 31 (A Coruña, Spain, 2023), p. Abstr 10614.
Reference: PAGE 33 (2025) Abstr 11336 [www.page-meeting.org/?abstract=11336]
Poster: Methodology - Other topics