Francisco Torres-Ruiz

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.

Modelling logistic growth by a new diffusion process: application to biological systems.

The present paper introduces a new diffusion process for the purpose of modelling logistic-type behaviour patterns. Unlike other processes in the same context, this one verifies that its mean function is a logistic curve. In addition, its transition density can be found explicitly, which allows to analyse inference from the discrete sampling of trajectories. The main features of the process will be analysed and the maximum likelihood estimation of parameters will be carried out through discrete sampling. Regarding the numerical problems found to solve the likelihood equations, several strategies are developed for obtaining initial solutions for the usual numerical procedures. Such strategies are compared by means of a simulation example. Also, another simulation study is carried out in order to compare the estimation in this process to that developed by means of continuous sampling in the logistic diffusion model considered by Giovanis and Skiadas (1999). Finally an example is given for the growth of a microorganism culture. This example illustrates the predictive possibilities of the new process, as well as its ability to study time variables formulated as first-passage-times.

Inference on a stochastic two-compartment model in tumor growth

A continuous-time model that incorporates several key elements in tumor dynamics is analyzed. More precisely, the form of proliferating and quiescent cell lines comes out from their relations with the whole tumor mass, giving rise to a two-dimensional diffusion process, generally time non-homogeneous. This model is able to include the effects of the mutual interactions between the two subpopulations. Estimation of the rates of the two subpopulations based on some characteristics of the involved diffusion processes is discussed when longitudinal data are available. To this aim, two procedures are presented. Some simulation results are developed in order to show the validity of these procedures as well as to compare them. An application to real data is finally presented.

A diffusion process to model generalized von Bertalanffy growth patterns: fitting to real data.

The von Bertalanffy growth curve has been commonly used for modeling animal growth (particularly fish). Both deterministic and stochastic models exist in association with this curve, the latter allowing for the inclusion of fluctuations or disturbances that might exist in the system under consideration which are not always quantifiable or may even be unknown. This curve is mainly used for modeling the length variable whereas a generalized version, including a new parameter b > or = 1, allows for modeling both length and weight for some animal species in both isometric (b = 3) and allometric (b not = 3) situations. In this paper a stochastic model related to the generalized von Bertalanffy growth curve is proposed. This model allows to investigate the time evolution of growth variables associated both with individual behaviors and mean population behavior. Also, with the purpose of fitting the above-mentioned model to real data and so be able to forecast and analyze particular characteristics, we study the maximum likelihood estimation of the parameters of the model. In addition, and regarding the numerical problems posed by solving the likelihood equations, a strategy is developed for obtaining initial solutions for the usual numerical procedures. Such strategy is validated by means of simulated examples. Finally, an application to real data of mean weight of swordfish is presented.

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.

Estimating the parameters of a Gompertz-type diffusion process by means of Simulated Annealing

Abstract This paper explores the application of the Simulated Annealing algorithm for the maximum likelihood estimation of the parameters of a Gompertz-type process. Firstly, the solution space is bounded using relevant information about the process provided by the sample data. Secondly, a proposal for improvement is made, namely the application of a second cycle of the algorithm, including a refinement factor. Finally, both the specifications for the application of the algorithm and the proposed improvement are validated through their application to simulated and real data.

A stochastic model related to the Richards-type growth curve. Estimation by means of simulated annealing and variable neighborhood search

For a Richards-type curve a diffusion process is constructed.The distribution and main characteristics of the process are analyzed.The estimation of the parameters is addressed by maximum likelihood.A procedure for bounding the parametric space is established.VND and SA algorithms are used for estimating the parameters. A stochastic diffusion model related to a reformulation of the Richards growth curve is proposed. The main characteristics of the process are studied, and the problem of maximum likelihood estimation for the parameters of the process is considered. Since a complex system of equations appears whose solution cannot be guaranteed via the classic numerical procedures, we suggest the use of metaheuristic optimization algorithms such as simulated annealing and variable neighborhood search. Given that the space of solutions is continuous and unbounded, some strategies are suggested for bounding it, and a description is provided for the application of the selected algorithms. In the case of the variable neighborhood search algorithm, a hybrid method is proposed in which it is combined with simulated annealing. Some examples based on simulated sample paths are developed in order to test the validity of the bounding method for the space of solutions, and a comparison is made between the application of both methods.

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.

Inferring the effect of therapies on tumor growth by using diffusion processes

Modeling the effect of therapies in cancer animal models remains a challenge. This point may be addressed by considering a diffusion process that models the tumor growth and a modified process that includes, in its infinitesimal mean, a time function modeling the effect of the therapy. In the case of a Gompertz diffusion process, where a control group and one or more treated groups are examined, a methodology to estimate this function has been proposed by Albano et al. (2011). This method has been applied to infer the effect of cisplatin and doxorubicin+cyclophosphamide on breast cancer xenografts. Although this methodology can be extended to other diffusion processes, it has an important restriction: it is necessary that a known diffusion process adequately fits the control group. Here, we propose the use of a stochastic process for a hypothetical control group, in such a way that both the control and the treated groups can be modeled by modified processes of the former. Thus, the comparison between models would allow estimating the real effect of the therapy. The new methodology has been validated by inferring the effects in breast cancer models, and we have checked the robustness of the procedure against the choice of stochastic model for the hypothetical control group. Finally, we have also applied the methodology to infer the effect of a therapeutic peptide and ovariectomy on the growth of a breast cancer xenograft, and its efficiency in modeling the effect of different treatments in the absence of control group data is shown.

Estimating a non-homogeneous Gompertz process with jumps as model of tumor dynamics

A non-homogeneous stochastic model based on a Gompertz-type diffusion process with jumps is proposed to describe the evolution of a solid tumor subject to an intermittent therapeutic program. Each therapeutic application, represented by a jump in the process, instantly reduces the tumor size to a fixed value and, simultaneously, increases the growth rate of the model to represent the toxicity of the therapy. This effect is described by introducing a time-dependent function in the drift of the process. The resulting model is a combination of several non-homogeneous diffusion processes characterized by different drifts, whose transition probability density function and main characteristics are studied. The study of the model is performed by distinguishing whether the therapeutic instances are fixed in advance or guided by a strategy based on the mean of the first-passage-time through a control threshold. Simulation studies are carried out for different choices of the parameters and time-dependent functions involved.