Stability analysis of a steady state of a model describing Alzheimer's disease and interactions with prion proteins.

Alzheimer's disease (AD) is a neuro-degenerative disease affecting more than 46 million people worldwide in 2015. AD is in part caused by the accumulation of A[Formula: see text] peptides inside the brain. These can aggregate to form insoluble oligomers or fibrils. Oligomers have the capacity to interact with neurons via membrane receptors such as prion proteins ([Formula: see text]). This interaction leads [Formula: see text] to be misfolded in oligomeric prion proteins ([Formula: see text]), transmitting a death signal to neurons. In the present work, we aim to describe the dynamics of A[Formula: see text] assemblies and the accumulation of toxic oligomeric species in the brain, by bringing together the fibrillation pathway of A[Formula: see text] peptides in one hand, and in the other hand A[Formula: see text] oligomerization process and their interaction with cellular prions, which has been reported to be involved in a cell-death signal transduction. The model is based on Becker-Döring equations for the polymerization process, with delayed differential equations accounting for structural rearrangement of the different reactants. We analyse the well-posedness of the model and show existence, uniqueness and non-negativity of solutions. Moreover, we demonstrate that this model admits a non-trivial steady state, which is found to be globally stable thanks to a Lyapunov function. We finally present numerical simulations and discuss the impact of model parameters on the whole dynamics, which could constitute the main targets for pharmaceutical industry.

Estimates and impact of lymphocyte division parameters from CFSE data using mathematical modelling.

Carboxyfluorescein diacetate succinimidyl ester (CFSE) labelling has been widely used to track and study cell proliferation. Here we use mathematical modelling to describe the kinetics of immune cell proliferation after an in vitro polyclonal stimulation tracked with CFSE. This approach allows us to estimate a set of key parameters, including ones related to cell death and proliferation. We develop a three-phase model that distinguishes a latency phase, accounting for non-divided cell behaviour, a resting phase and the active phase of the division process. Parameter estimates are derived from model results, and numerical simulations are then compared to the dynamics of in vitro experiments, with different biological assumptions tested. Our model allows us to compare the dynamics of CD4+ and CD8+ cells, and to highlight their kinetic differences. Finally we perform a sensitivity analysis to quantify the impact of each parameter on proliferation kinetics. Interestingly, we find that parameter sensitivity varies with time and with cell generation. Our approach can help biologists to understand cell proliferation mechanisms and to identify potential pathological division processes.

Analysis of temozolomide resistance in low-grade gliomas using a mechanistic mathematical model.

Understanding how tumors develop resistance to chemotherapy is a major issue in oncology. When treated with temozolomide (TMZ), an oral alkylating chemotherapy drug, most low-grade gliomas (LGG) show an initial volume decrease but this effect is rarely long lasting. In addition, it has been suggested that TMZ may drive tumor progression in a subset of patients as a result of acquired resistance. Using longitudinal tumor size measurements from 121 patients, the aim of this study was to develop a semi-mechanistic mathematical model to determine whether resistance of LGG to TMZ was more likely to result from primary and/or from chemotherapy-induced acquired resistance that may contribute to tumor progression. We applied the model to a series of patients treated upfront with TMZ (n = 109) or PCV (procarbazine, CCNU, vincristine) chemotherapy (n = 12) and used a population mixture approach to classify patients according to the mechanism of resistance most likely to explain individual tumor growth dynamics. Our modeling results predicted acquired resistance in 51% of LGG treated with TMZ. In agreement with the different biological effects of nitrosoureas, none of the patients treated with PCV were classified in the acquired resistance group. Consistent with the mutational analysis of recurrent LGG, analysis of growth dynamics using mathematical modeling suggested that in a subset of patients, TMZ might paradoxically contribute to tumor progression as a result of chemotherapy-induced resistance. Identification of patients at risk of developing acquired resistance is warranted to better define the role of TMZ in LGG.

Increasing the Time Interval between PCV Chemotherapy Cycles as a Strategy to Improve Duration of Response in Low-Grade Gliomas: Results from a Model-Based Clinical Trial Simulation.

BACKGROUND: We previously developed a mathematical model capturing tumor size dynamics of adult low-grade gliomas (LGGs) before and after treatment either with PCV (Procarbazine, CCNU, and Vincristine) chemotherapy alone or with radiotherapy (RT) alone. OBJECTIVE: The aim of the present study was to present how the model could be used as a simulation tool to suggest more effective therapeutic strategies in LGGs. Simulations were performed to identify schedule modifications that might improve PCV chemotherapy efficacy. METHODS: Virtual populations of LGG patients were generated on the basis of previously evaluated parameter distributions. Monte Carlo simulations were performed to compare treatment efficacy across in silico clinical trials. RESULTS: Simulations predicted that RT plus PCV would be more effective in terms of duration of response than RT alone. Additional simulations suggested that, in patients treated with PCV chemotherapy, increasing the interval between treatment cycles up to 6 months from the standard 6 weeks can increase treatment efficacy. The predicted median duration of response was 4.3 years in LGGs treated with PCV cycles given every 6 months versus 3.1 years in patients treated with the classical regimen. CONCLUSION: The present study suggests that, in LGGs, mathematical modeling could facilitate clinical research by helping to identify, in silico, potentially more effective therapeutic strategies.