A mathematical model to study the effects of drugs administration on tumor growth dynamics

Abstract

A mathematical model for describing the cancer growth dynamics in response to anticancer agents administration in xenograft models is discussed. The model consists of a system of ordinary differential equations involving five parameters (three for describing the untreated growth and two for describing the drug action). Tumor growth in untreated animals is modelled by an exponential growth followed by a linear growth. In treated animals, tumor growth rate is decreased by an additional factor proportional to both drug concentration and proliferating cells. The mathematical analysis conducted in this paper highlights several interesting properties of this tumor growth model. It suggests also effective strategies to design in vivo experiments in animals with potential saving of time and resources. For example, the drug concentration threshold for the tumor eradication, the delay between drug administration and tumor regression, and a time index that measures the efficacy of a treatment are derived and discussed. The model has already been employed in several drug discovery projects. Its application on a data set coming from one of these projects is discussed in this paper.

Introduction

The in vivo evaluation of the effects of an antitumor drug is an important step in preclinical development of oncology drugs. For this purpose, several experiments are usually performed, in which human tumor cells from immortalized cell lines are inoculated into athymic mice. Some of these animals are then treated with antitumor drugs and the tumor volumes are measured at different times in both treated and untreated mice. The ability of the drug to inhibit tumor growth is usually measured comparing the average tumor weights in treated and control animals at the end of experiment, thus allowing the comparison and ranking of different compounds or different dosages/schedules of the same compound. As discussed in [1], [2], [3] the growth dynamics of inoculated cells can be considered a good model of the tumor growth dynamics in human.

Building a mathematical model of the tumor growth that assesses drug efficacy on the basis of the whole time series of tumor weights instead of adopting a criterion simply based on a single measurement at the end of the treatment is a challenging task. At the same time, a model able to correctly predict experimental data (without actually performing the experiment) would be of great value in order to save time and money in the drug discovery process. To be of practical use, such a model should be able to: (i) describe the observed tumor growth curves using few physically meaningful parameters, identifiable from experimental data; (ii) make predictions of tumor growth kinetics in response to different treatments (i.e. dosages/schedules); (iii) assess drug potency (independently of the dose levels and schedules).

Over the past two decades, in vivo growth kinetics of solid tumors has been extensively studied and several mathematical models have appeared in the literature. Tumor growth is a complicated phenomenon involving several correlated processes and the existing studies reflect different paradigms, ranging from simple and empirical models to more complex functional ones. The former use empirical mathematical equations (e.g. sigmoidal functions such as logistic, Verhulst, Gompertz, and von Bertalanffy) to describe the growth curve of macroscopic variables such as volume, mass or size of cellular population [4], [5]. Their aim is to model the tumor growth even without a mechanistic description of the underlying physiological processes. A drawback of this class of models is that it is not straightforward to predict the modification of the growth curve in response to changes in the drug administration schedule. Functional models, conversely, are based on a set of assumptions about biological growth reflecting a microscopic viewpoint, involving cell cycle kinetics and/or cell–cell interactions [6], [7]. Such models usually represent the cell population in its heterogeneity; in the simplest case the whole population consists of two subpopulations only: the proliferating and the quiescent [8]. More complex models describe population as age-structured and take into account more than two subpopulations related to specific phases of the mitotic cell cycle. In some cases, they take into account also the spatial disposition of tumor cells. These models, based on well-understood biological principles, are generally complex with a much larger number of parameters compared to empirical models [9], [10], [11], [12], [13]. As a consequence, it is generally difficult to fit functional models versus experimental data since overparametrization can be avoided only if further ‘microscopic’ observations are available (e.g. flow cytometry, PET, etc.) [14], [15].

This paper analyzes a new mathematical model, identifiable from experiments performed in nude mice as part of the typical drug research and development project, able to assess drug potency and predict tumor growth in response of different treatments.