Stochastic Gompertz model of tumour cell growth

In this communication, based upon the deterministic Gompertz law of cell growth, a stochastic model in tumour growth is proposed. This model takes account of both cell fission and mortality too. The corresponding density function of the size of the tumour cells obeys a functional Fokker--Planck equation which can be solved analytically. It is found that the density function exhibits an interesting "multi-peak" structure generated by cell fission as time evolves. Within this framework the action of therapy is also examined by simply incorporating a therapy term into the deterministic cell growth term.     

Persistent random motion: Uncovering cell migration dynamics

In this paper we study analytically the stick-slip models recently introduced to explain the stochastic migration of free cells. We show that persistent motion of cells of many different types is compatible with stochastic reorientation models which admit an analytical mesoscopic treatment. This is proved by examining and discussing experimental data compiled from different sources in the literature, and by fitting some of these results too. We are able to explain many of the ‘apparently complex’ migration patterns obtained recently from cell tracking data, like power-law dependences in the mean square displacement or non-Gaussian behavior for the kurtosis and the velocity distributions, which depart from the predictions of the classical Ornstein-Uhlenbeck process.     

phylosem: A fast and simple R package for phylogenetic inference and trait imputation using phylogenetic structural equation models

Phylogenetic comparative methods (PCMs) can be used to study evolutionary relationships and trade-offs among species traits. Analysts using PCM may want to (1) include latent variables, (2) estimate complex trait interdependencies, (3) predict missing trait values, (4) condition predicted traits upon phylogenetic correlations and (5) estimate relationships as slope parameters that can be compared with alternative regression methods. The Comprehensive R Archive Network (CRAN) includes well-documented software for phylogenetic linear models (phylolm), phylogenetic path analysis (phylopath), phylogenetic trait imputation (Rphylopars) and structural equation models (sem), but none of these can simultaneously accomplish all five analytical goals. We therefore introduce a new package phylosem for phylogenetic structural equation models (PSEM) and summarize features and interface. We also describe new analytical options, where users can specify any combination of Ornstein-Uhlenbeck, Pagel's-δ and Pagel's-λ transformations for species covariance. For the first time, we show that PSEM exactly reproduces estimates (and standard errors) for simplified cases that are feasible in sem, phylopath, phylolm and Rphylopars and demonstrate the approach by replicating a well-known case study involving trade-offs in plant energy budgets.     

Correlated evolution of diaspore traits and potential frugivore-mediated selection in a fleshy-fruited tropical lineage (Artabotrys,Annonaceae)

Dispersal syndromes are often defined by reference to fruit traits that are associated with distinct frugivore guilds. Studies rarely examine the relationship between seed traits and frugivores or test the alternative hypothesis that traits are shaped by climatic variables. We assess whether the evolution of seed size and physical defense are correlated with dispersal-related traits and climatic variables in Artabotrys, a fleshy-fruited tropical lineage. Diaspore traits and WorldClim bioclimatic variables were compiled for 43 species. Correlated evolution was evaluated using phylogenetic regression and model-fitting approaches. The best-fitting multioptima Ornstein-Uhlenbeck model suggests that lineages with smooth testa and thin pericarp (SP) have evolved toward smaller seeds with a thinner testa, whereas lineages with rough testa and/or thick pericarp have evolved toward larger seeds with a thicker testa. A smooth testa facilitates spitting and/or swallowing of intact seeds while fruits with thin pericarp may be preferentially consumed by frugivores with less destructive oral processing, enabling lower investment in seed physical defense in SP lineages. Moreover, small seeds are more likely to be swallowed intact with a food bolus. The effect of climate on seed size and physical defense is equivocal and warrants further investigation.     

A stochastic model in tumor growth

A stochastic model of solid tumor growth based on deterministic Gompertz law is presented. Tumor cells evolution is described by a one-dimensional diffusion process limited by two absorbing boundaries representing healing threshold and patient death (carrying capacity), respectively. Via a numerical approach the first exit time problem is analysed for the process inside the region restricted by the boundaries. The proposed model is also implemented to simulate the effects of a time-dependent therapy. Finally, some numerical results are obtained for the specific case of a parathyroid tumor.     

First passage time problem for a drifted Ornstein-Uhlenbeck process

We consider a continuous stochastic process defined as a drifted Ornstein-Uhlenbeck, for which the first passage time is of interest. The process being non-homogeneous, the first passage time probability density function cannot be found analytically, but numerical methods enable to find its estimate. Estimating the first passage time implies solving an unsteady convection-diffusion equation, with variable coefficients, and we use an implicit Euler scheme to solve it. This work is applied to simulated data, and the continuous process is inspired from recent work on biological marker modelling for HIV-positive patients. The first passage time probability density function can be useful to compare the marker progression in different groups. Numerical results show that the first passage time is highly dependent from the process perturbation, and is then more relevant than methods not considering the stochastic process directly to compare the progression.     

On the Classification of Experimental Data Modeled Via a Stochastic Leaky Integrate and Fire Model Through Boundary Values

We present a computational algorithm aimed to classify single unit spike trains on the basis of observed interspikes intervals (ISI). The neuronal activity is modeled with a stochastic leaky integrate and fire model and the inverse first passage time method is extended to the Ornstein-Uhlenbeck (OU) process. Differences between spike trains are detected in terms of the boundary shape. The proposed classification method is applied to the analysis of multiple single units recorded simultaneously in the thalamus and in the cerebral cortex of unanesthetized rats during spontaneous activity. We show the existence of at least three different firing patterns that could not be classified using the usual statistical indices. PACS: 87.19.La MSC: 60K30, 60J60, 65C40, 62P10     

Optimum signal in a diffusion leaky integrate-and-fire neuronal model

An optimum signal in the Ornstein-Uhlenbeck neuronal model is determined on the basis of interspike interval data. Two criteria are proposed for this purpose. The first, the classical one, is based on searching for maxima of the slope of the frequency transfer function. The second one uses maximum of the Fisher information, which is, under certain conditions, the inverse variance of the best possible estimator. The Fisher information is further normalized with respect to the time required to make the observation on which the signal estimation is performed. Three variants of the model are investigated. Beside the basic one, we use the version obtained by inclusion of the refractory period. Finally, we investigate such a version of the model in which signal and the input parameter of the model are in a nonlinear relationship. The results show that despite qualitative similarity between the criteria, there is substantial quantitative difference. As a common feature, we found that in the Ornstein-Uhlenbeck model with increasing noise the optimum signal decreases and the coding range gets broader.