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Glioblastoma multiforme (GBM) is one of the most deadly forms of cancer. Methods of characterizing these tumours are valuable for improving predictions of their progression and response to treatment. A mathematical model called the proliferation-invasion (PI) model has been used extensively in the literature to model the growth of these tumours, though it relies on known values of two key parameters: the tumour cell diffusivity and proliferation rate. Unfortunately, these parameters are difficult to estimate in a patient-specific manner, making personalized tumour forecasting challenging. In this paper, we develop and apply a deep learning model capable of making accurate estimates of these key GBM-characterizing parameters while simultaneously producing a full prediction of the tumour progression curve. Our method uses two sets of multi sequence MRI in order to produce estimations and relies on a preprocessing pipeline which includes brain tumour segmentation and conversion to tumour cellularity. We first apply our deep learning model to synthetic tumours to showcase the model's capabilities and identify situations where prediction errors are likely to occur. We then apply our model to a clinical dataset consisting of five patients diagnosed with GBM. For all patients, we derive evidence-based estimates for each of the PI model parameters and predictions for the future progression of the tumour, along with estimates of the parameter uncertainties. Our work provides a new, easily generalizable method for the estimation of patient-specific tumour parameters, which can be built upon to aid physicians in designing personalized treatments. • Mathematical models are used to predict tumour growth and response to treatment. • Use of models in the clinic requires patient-specific tumour characterizations. • Deep learning algorithms can use MRIs to characterize tumours and inform models. • Deep learning characterization models show advantages over traditional methods. [ABSTRACT FROM AUTHOR]

In this paper, I present an analytical framework for polymorphic evolutionary games suitable for explicitly modeling evolutionary processes in diploid populations with sexual reproduction. The principal aspect of the proposed approach is adding diploid genetics cum sexual recombination to a traditional evolutionary game, and switching from phenotypes to haplotypes as the new game׳s pure strategies. Here, the relevant pure strategy׳s payoffs derived by summing the payoffs of all the phenotypes capable of producing gametes containing that particular haplotype weighted by the pertinent probabilities. The resulting game is structurally identical to the familiar Evolutionary Games with non-linear pure strategy payoffs ( Hofbauer and Sigmund, 1998 . Cambridge University Press), and can be analyzed in terms of an established analytical framework for such games. And these results can be translated into the terms of genotypic, and whence, phenotypic evolutionary stability pertinent to the original game. [ABSTRACT FROM AUTHOR]

We investigate the dynamics and transitions to extinction of hypercycles governed by periodic orbits. For a large enough number of hypercycle species ( n > 4 ) the existence of a stable periodic orbit has been previously described, showing an apparent coincidence of the vanishing of the periodic orbit with the value of the replication quality factor Q where two unstable (non-zero) equilibrium points collide (named Q SS ). It has also been reported that, for values below Q SS , the system goes to extinction. In this paper, we use a suitable Poincaré map associated to the hypercycle system to analyze the dynamics in the bistability regime, where both oscillatory dynamics and extinction are possible. The stable periodic orbit is identified, together with an unstable periodic orbit. In particular, we are able to unveil the vanishing mechanism of the oscillatory dynamics: a saddle-node bifurcation of periodic orbits as the replication quality factor, Q , undergoes a critical fidelity threshold, Q PO . The identified bifurcation involves the asymptotic extinction of all hypercycle members, since the attractor placed at the origin becomes globally stable for values Q < Q PO . Near the bifurcation, these extinction dynamics display a periodic remnant that provides the system with an oscillating delayed transition. Surprisingly, we found that the value of Q PO is slightly higher than Q SS , thus identifying a gap in the parameter space where the oscillatory dynamics has vanished while the unstable equilibrium points are still present. We also identified a degenerate bifurcation of the unstable periodic orbits for Q =1. [ABSTRACT FROM AUTHOR]