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.

[1]  L. de Ridder,et al.  Autologous spheroid culture: a screening tool for human brain tumour invasion. , 2000, Critical reviews in oncology/hematology.

[2]  D A Weitz,et al.  Glioma expansion in collagen I matrices: analyzing collagen concentration-dependent growth and motility patterns. , 2005, Biophysical journal.

[3]  Alissa M. Weaver,et al.  Tumor Morphology and Phenotypic Evolution Driven by Selective Pressure from the Microenvironment , 2006, Cell.

[4]  Nicolas André,et al.  Computational oncology — mathematical modelling of drug regimens for precision medicine , 2016, Nature Reviews Clinical Oncology.

[5]  Maciej Swat,et al.  Systems oncology: towards patient-specific treatment regimes informed by multiscale mathematical modelling. , 2015, Seminars in cancer biology.

[6]  L. Puricelli,et al.  Modulation of fibronectin expression and proteolytic activity associated with the invasive and metastatic phenotype in two new murine mammary tumor cell lines. , 1997, International journal of oncology.

[7]  Luigi Preziosi,et al.  Cancer Modelling and Simulation , 2003 .

[8]  T. Mesti,et al.  Malignant gliomas: old and new systemic treatment approaches , 2016, Radiology and oncology.

[9]  H. Aarstad,et al.  Co‐culture of Head and Neck Squamous Cell Carcinoma Spheroids with Autologous Monocytes Predicts Prognosis , 2008, Scandinavian journal of immunology.

[10]  E. Denisov,et al.  Cancer Invasion: Patterns and Mechanisms , 2015, Acta naturae.

[11]  D. Altman,et al.  Applying the right statistics: analyses of measurement studies , 2003, Ultrasound in obstetrics & gynecology : the official journal of the International Society of Ultrasound in Obstetrics and Gynecology.

[12]  Dietmar W Hutmacher,et al.  A multiscale road map of cancer spheroids – incorporating experimental and mathematical modelling to understand cancer progression , 2013, Journal of Cell Science.

[13]  H M Byrne,et al.  Growth of confined cancer spheroids: a combined experimental and mathematical modelling approach. , 2013, Integrative biology : quantitative biosciences from nano to macro.

[14]  Martin Fussenegger,et al.  Method for generation of homogeneous multicellular tumor spheroids applicable to a wide variety of cell types. , 2003, Biotechnology and bioengineering.

[15]  Andrew G. Clark,et al.  Modes of cancer cell invasion and the role of the microenvironment. , 2015, Current opinion in cell biology.

[16]  Wei Sun,et al.  Three-dimensional in vitro cancer models: a short review , 2014, Biofabrication.

[17]  Mark A. J. Chaplain,et al.  Integrating Intracellular Dynamics Using CompuCell3D and Bionetsolver: Applications to Multiscale Modelling of Cancer Cell Growth and Invasion , 2012, PloS one.

[18]  H. Kleinman,et al.  In Vitro Microtumors Provide a Physiologically Predictive Tool for Breast Cancer Therapeutic Screening , 2015, PloS one.

[19]  Philipp M. Altrock,et al.  The mathematics of cancer: integrating quantitative models , 2015, Nature Reviews Cancer.

[20]  Gabor Forgacs,et al.  The interplay of cell-cell and cell-matrix interactions in the invasive properties of brain tumors. , 2006, Biophysical journal.

[21]  D. Altman,et al.  STATISTICAL METHODS FOR ASSESSING AGREEMENT BETWEEN TWO METHODS OF CLINICAL MEASUREMENT , 1986, The Lancet.

[22]  Kristin R. Swanson,et al.  Patient-Specific Mathematical Neuro-Oncology: Using a Simple Proliferation and Invasion Tumor Model to Inform Clinical Practice , 2015, Bulletin of mathematical biology.

[23]  L. weiswald,et al.  Spherical Cancer Models in Tumor Biology1 , 2015, Neoplasia.

[24]  A. Beckett,et al.  AKUFO AND IBARAPA. , 1965, Lancet.

[25]  Michael Berens,et al.  A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment. , 2007, Biophysical journal.

[26]  A. Guzman,et al.  The effect of fibrillar matrix architecture on tumor cell invasion of physically challenging environments. , 2014, Biomaterials.

[27]  C. Suárez,et al.  Mathematical Modeling of Human Glioma Growth Based on Brain Topological Structures: Study of Two Clinical Cases , 2012, PloS one.

[28]  Leonard M Sander,et al.  Estimating the cell density and invasive radius of three-dimensional glioblastoma tumor spheroids grown in vitro. , 2007, Applied optics.