Estimating and determining the effect of a therapy on tumor dynamics by means of a modified Gompertz diffusion process Giuseppina Albano a,n , Virginia Giorno b , Patricia Román-Román c , Sergio Román-Román d , Francisco Torres-Ruiz c a Dip. di Scienze Economiche e Statistiche, Università di Salerno, Italy b Dip. di Studi e Ricerche Aziendali (Management and Information Technology), Università di Salerno, Italy c Dpto. de Estadística e Investigación Operativa, Universidad de Granada, Spain d Dép. de Recherche Translationnelle, Institut Curie, France HIGHLIGHTS  Methodology to estimate the functions representing the effect of a therapy.  Criteria for obtaining insights into the nature of a therapy in experimental studies.  Application to real data of a patient-derived uveal melanoma xenografted in mice. article info Article history: Received 17 October 2013 Received in revised form 8 September 2014 Accepted 9 September 2014 Available online 19 September 2014 Keywords: Tumor growth Therapy effect estimation Prevalent effect Kullback–Leibler divergence Resistor average distance abstract 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. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction In the last decades, increasing attention has been paid to the formulation and analysis of mathematical procedures for modeling tumor growth and the effect of therapies in cancer animal models. Innovative and informative analytical methods modeling tumor growth in vivo could result in more accurate interpretation of data obtained from these animal models, and help pharmacologists to adapt administration schedules of drugs at the preclinical setting. Moreover such tools could also potentially give insights into the mechanism of action of the drugs in vivo. Actually, mechanistic studies of drugs are conducted in vitro and both the host and the tumor microenvironment can dramatically affect the mechanism by which a given drug or drug combination is displaying its activity in vivo. In a first approach, the models are related to a particular growth curve that is a solution of a differential equation. The deterministic models more commonly used in the study of tumor growth are the Malthusian (associated with the exponential curve) and those related to sigmoidal curves as the logistic or Gompertz. This second model is the most widely accepted to capture dynamics in solid tumors (see the classic work of Norton, 1988, or that of Gerlee, 2013 for a recent historical perspective). In order to take into consideration the environmental fluctuations that are often cause of discrepancies between preclinical data and theoretical predictions, the notion of growth in random environment has been formulated (see, for instance, Nobile and Ricciardi, 1980 and references therein). Various approaches (deterministics and stochastics), based on the population dynamics, are proposed in the literature to analyze Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/yjtbi Journal of Theoretical Biology http://dx.doi.org/10.1016/j.jtbi.2014.09.014 0022-5193/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: +39 089962645. E-mail address: pialbano@unisa.it (G. Albano). Journal of Theoretical Biology 364 (2015) 206–219