OpenEP: an open-source simulator for electroporation-based tumor treatments.

Electroporation (EP), the increase of cell membrane permeability due to the application of electric pulses, is a universal phenomenon with a broad range of applications. In medicine, some of the foremost EP-based tumor treatments are electrochemotherapy (ECT), irreversible electroporation, and gene electrotransfer (GET). The electroporation phenomenon is explained as the formation of cell membrane pores when a transmembrane cell voltage reaches a threshold value. Predicting the outcome of an EP-based tumor treatment consists of finding the electric field distribution with an electric threshold value covering the tumor (electroporated tissue). Threshold and electroporated tissue are also a function of the number of pulses, constituting a complex phenomenon requiring mathematical modeling. We present OpenEP, an open-source specific purpose simulator for EP-based tumor treatments, modeling among other variables, threshold, and electroporated tissue variations in time. Distributed under a free/libre user license, OpenEP allows the customization of tissue type; electrode geometry and material; pulse type, intensity, length, and frequency. OpenEP facilitates the prediction of an optimal EP-based protocol, such as ECT or GET, defined as the critical pulse dosage yielding maximum electroporated tissue with minimal damage. OpenEP displays a highly efficient shared memory implementation by taking advantage of parallel resources; this permits a rapid prediction of optimal EP-based treatment efficiency by pulse number tuning.

Microenvironmental influence on microtumour infiltration patterns: 3D-mathematical modelling supported by in vitro studies.

Mathematical modelling approaches have become increasingly abundant in cancer research. Tumour infiltration extent and its spatial organization depend both on the tumour type and stage and on the bio-physicochemical characteristics of the microenvironment. This sets a complex scenario that often requires a multidisciplinary and individually adjusted approach. The ultimate goal of this work is to present an experimental/numerical combined method for the development of a three-dimensional mathematical model with the ability to reproduce the growth and infiltration patterns of a given avascular microtumour in response to different microenvironmental conditions. The model is based on a diffusion-convection reaction equation that considers logistic proliferation, volumetric growth, a rim of proliferative cells at the tumour surface, and invasion with diffusive and convective components. The parameter values of the model were fitted to experimental results while radial velocity and diffusion coefficients were made spatially variable in a case-specific way through the introduction of a shape function and a diffusion-limited-aggregation (DLA)-derived fractal matrix, respectively, according to the infiltration pattern observed. The in vitro model consists of multicellular tumour spheroids (MTSs) of an epithelial mammary tumour cell line (LM3) immersed in a collagen I gel matrix with a standard culture medium ("naive" matrix) or a conditioned medium from adipocytes or preadipocytes ("conditioned" matrix). It was experimentally determined that both adipocyte and preadipocyte conditioned media had the ability to change the MTS infiltration pattern from collective and laminar to an individual and atomized one. Numerical simulations were able to adequately reproduce qualitatively and quantitatively both kinds of infiltration patterns, which were determined by area quantification, analysis of fractal dimensions and lacunarity, and Bland-Altman analysis. These results suggest that the combined approach presented here could be established as a new framework with interesting potential applications at both the basic and clinical levels in the oncology area.

Mathematical modelling of microtumour infiltration based on in vitro experiments.

The present mathematical models of microtumours consider, in general, volumetric growth and spherical tumour invasion shapes. Nevertheless in many cases, such as in gliomas, a need for more accurate delineation of tumour infiltration areas in a patient-specific manner has arisen. The objective of this study was to build a mathematical model able to describe in a case-specific way as well as to predict in a probabilistic way the growth and the real invasion pattern of multicellular tumour spheroids (in vitro model of an avascular microtumour) immersed in a collagen matrix. The two-dimensional theoretical model was represented by a reaction-convection-diffusion equation that considers logistic proliferation, volumetric growth, a rim with proliferative cells at the tumour surface and invasion with diffusive and convective components. Population parameter values of the model were extracted from the experimental dataset and a shape function that describes the invasion area was derived from each experimental case by image processing. New possible and aleatory shape functions were generated by data mining and Monte Carlo tools by means of a satellite EGARCH model, which were fed with all the shape functions of the dataset. Then the main model is used in two different ways: to reproduce the growth and invasion of a given experimental tumour in a case-specific manner when fed with the corresponding shape function (descriptive simulations) or to generate new possible tumour cases that respond to the general population pattern when fed with an aleatory-generated shape function (predictive simulations). Both types of simulations are in good agreement with empirical data, as it was revealed by area quantification and Bland-Altman analysis. This kind of experimental-numerical interaction has wide application potential in designing new strategies able to predict as much as possible the invasive behaviour of a tumour based on its particular characteristics and microenvironment.