Exact solutions and asymptotic behaviors for the reflected Wiener, Ornstein-Uhlenbeck and Feller diffusion processes.

We analyze the transition probability density functions in the presence of a zero-flux condition in the zero-state and their asymptotic behaviors for the Wiener, Ornstein Uhlenbeck and Feller diffusion processes. Particular attention is paid to the time-inhomogeneous proportional cases and to the time-homogeneous cases. A detailed study of the moments of first-passage time and of their asymptotic behaviors is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions for the restricted Wiener, Ornstein-Uhlenbeck and Feller processes are proved. Specific applications of the results to queueing systems are provided.

Inference on the effect of non homogeneous inputs in Ornstein-Uhlenbeck neuronal modeling.

A non-homogeneous Ornstein-Uhlembeck (OU) diffusion process is considered as a model for the membrane potential activity of a single neuron. We assume that, in the absence of stimuli, the neuron activity is described via a time-homogeneous process with linear drift and constant infinitesimal variance. When a sequence of inhibitory and excitatory post-synaptic potentials occurres with generally time-dependent rates, the membrane potential is then modeled by means of a non-homogeneous OU-type process. From a biological point of view it becomes important to understand the behavior of the membrane potential in the presence of such stimuli. This issue means, from a statistical point of view, to make inference on the resulting process modeling the phenomenon. To this aim, we derive some probabilistic properties of a non-homogeneous OU-type process and we provide a statistical procedure to fit the constant parameters and the time-dependent functions involved in the model. The proposed methodology is based on two steps: the first one is able to estimate the constant parameters, while the second one fits the non-homogeneous terms of the process. Related to the second step two methods are considered. Some numerical evaluations based on simulation studies are performed to validate and to compare the proposed procedures.

Estimating and determining the effect of a therapy on tumor dynamics by means of a modified Gompertz diffusion process.

A modified Gompertz diffusion process is considered to model tumor dynamics. The infinitesimal mean of this process includes non-homogeneous terms describing the effect of therapy treatments able to modify the natural growth rate of the process. Specifically, therapies with an effect on cell growth and/or cell death are assumed to modify the birth and death parameters of the process. This paper proposes a methodology to estimate the time-dependent functions representing the effect of a therapy when one of the functions is known or can be previously estimated. This is the case of therapies that are jointly applied, when experimental data are available from either an untreated control group or from groups treated with single and combined therapies. Moreover, this procedure allows us to establish the nature (or, at least, the prevalent effect) of a single therapy in vivo. To accomplish this, we suggest a criterion based on the Kullback-Leibler divergence (or relative entropy). Some simulation studies are performed and an application to real data is presented.

A stochastic model of cancer growth subject to an intermittent treatment with combined effects: reduction in tumor size and rise in growth rate.

A model of cancer growth based on the Gompertz stochastic process with jumps is proposed to analyze the effect of a therapeutic program that provides intermittent suppression of cancer cells. In this context, a jump represents an application of the therapy that shifts the cancer mass to a return state and it produces an increase in the growth rate of the cancer cells. For the resulting process, consisting in a combination of different Gompertz processes characterized by different growth parameters, the first passage time problem is considered. A strategy to select the inter-jump intervals is given so that the first passage time of the process through a constant boundary is as large as possible and the cancer size remains under this control threshold during the treatment. A computational analysis is performed for different choices of involved parameters. Finally, an estimation of parameters based on the maximum likelihood method is provided and some simulations are performed to illustrate the validity of the proposed procedure.

On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model.

An Ornstein-Uhlenbeck diffusion process is considered as a model for the membrane potential activity of a single neuron. We assume that the neuron is subject to a sequence of inhibitory and excitatory post-synaptic potentials that occur with time-dependent rates. The resulting process is characterized by time-dependent drift. For this model, we construct the return process describing the membrane potential. It is a non homogeneous Ornstein-Uhlenbeck process with jumps on which the effect of random refractoriness is introduced. An asymptotic analysis of the process modeling the number of firings and the distribution of interspike intervals is performed under the assumption of exponential distribution for the firing time. Some numerical evaluations are performed to provide quantitative information on the role of the parameters.

On the effect of a therapy able to modify both the growth rates in a Gompertz stochastic model.

A Gompertz-type diffusion process characterized by the presence of exogenous factors in the drift term is considered. Such a process is able to describe the dynamics of populations in which both the intrinsic rates are modified by means of time-dependent terms. In order to quantify the effect of such terms the evaluation of the relative entropy is made. The first passage time problem through suitable boundaries is studied. Moreover, some simulation results are shown in order to capture the dependence of the involved functions on the parameters. Finally, an application to tumor growth is presented and simulation results are shown.

On the therapy effect for a stochastic growth Gompertz-type model.

We consider a diffusion model based on a generalized Gompertz deterministic growth in which carrying capacity depends on the initial size of the population. The drift of the resulting process is then modified by introducing a time-dependent function, called "therapy", in order to model the effect of an exogenous factor. The transition probability density function and the related moments for the proposed process are obtained. A study of the influence of the therapy on several characteristics of the model is performed. The first-passage-time problem through time-dependent boundaries is also analyzed. Finally, an application to real data concerning a rabbit population subject to particular therapies is presented.

Inferring the effect of therapy on tumors showing stochastic Gompertzian growth.

The present work deals with a Gompertz-type diffusion process, which includes in the drift term a time-dependent function C(t) representing the effect of a therapy able to modify the dynamics of the underlying process. However, in experimental studies is not immediate to deduce the functional form of C(t) from a treatment protocol. So a statistical approach is proposed in order to estimate this function when a control group and one or more treated groups are observed. In order to validate the proposed strategy a simulation study for several interesting functional forms of C(t) has been carried out. Finally, an application to infer the net effect of cisplatin and doxorubicin+cyclophosphamide in actual murine models is presented.

A Wiener-type neuronal model in the presence of exponential refractoriness.

An instantaneous return process in the presence of random refractoriness for Wiener model of single neuron activity is considered. The case of exponential distributed refractoriness is analyzed and expressions for output distributions and interspike intervals density are obtained in closed form. A computational study is performed to elucidate the role played by the model parameters in affecting the firing probabilities and the interspike distribution.