Stochastic geometric models, and related statistical issues in tumour-induced angiogenesis Vincenzo Capasso * , Alessandra Micheletti, Daniela Morale Dipartimento di Matematica, Università degli Studi di Milano and CIMAB (InterUniversity Centre for Mathematics Applied to Environment, Biology, and Medicine), Via C. Saldini, 50, 20133 Milano, Italy article info Article history: Received 12 February 2008 Accepted 5 March 2008 Available online 6 April 2008 Keywords: Tumor angiogenesis Stochastic geometry Stochastic distributions Multiple scales abstract In the modelling and statistical analysis of tumor-driven angiogenesis it is of great importance to handle random closed sets of different (though integer) Hausdorff dimensions, usually smaller than the full dimension of the relevant space. Here an original approach is reported, based on random generalized densities (distributions) á la Dirac- Schwartz, and corresponding mean generalized densities. The above approach also suggests methods for the statistical estimation of geometric densities of the stochastic fibre system that characterize the morphology of a real vascular system. A quantitative description of the evolution of tumor-driven angiogenesis requires the mathematical modelling of a strongly coupled system of a stochastic branching-and-growth process of fibres, modelling the network of blood vessels, and a family of underlying fields, modelling biochemical signals. Methods for reducing complexity include homogenization at mesoscales, thus leading to hybrid models (deterministic at the larger scale, and stochastic at lower scales); in tumor-driven angiogenesis the two scales can be bridged by introducing a mesoscale at which one locally averages the microscopic branch- ing-and-growth process, in presence of a sufficiently large number of vessels (fibers). Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction The understanding of the principles and the dominant mechanisms underlying tumor growth is an essential prere- quisite for identifying optimal control strategies, in terms of prevention and treatment. Predictive mathematical models which are capable of producing quantitative morphological features of developing tumour and blood vessels can contrib- ute to this. The study of angiogenesis has such potential for providing new therapies that it has received enthusiastic interest from the phar- maceutical and biotechnology industries. Many of the compounds now under investigation inhibit angiogenesis and thus the growth of the cancer. Tumor growth develops thanks to the nutritional underlying field, driven by blood circulation (and thus enhanced by angiogenesis) [29]. On the other hand angiogenesis is activated by the presence of chemicals released by the tumor mass. Tumour- induced angiogenesis is believed to occur when normal tissue vas- culature is no longer able to support growth of an avascular tu- mour. At this stage the tumour cells, lacking nutrients and oxygen, become hypoxic. This is assumed to trigger cellular release of tumour angiogenic factors (TAF’s) which start to diffuse into the surrounding tissue and approach endothelial cells (EC’s) of nearby blood vessels [30]. EC’s subsequently respond to the TAF concen- tration gradients by forming sprouts, dividing and migrating to- wards the tumour. A summary of these mechanisms can be found in the recent paper by Carmeliet [36] (see also Figs. 1–3 where examples of real or simulated vascular networks are depicted). An important goal would then be the integration of mathemat- ical models for angiogenesis and tumor growth, but the existing unsolved complexity of the individual models has still prevented such an integration. A satisfactory mathematical modelling of angiogenesis and of many other fibre processes requires a theory of stochastic branch- ing-and-growth fibre processes, evolving in time, and strongly cou- pled with a set of underlying fields. A major difficulty derives from the strong coupling of the ki- netic parameters of the relevant branching-and-growth process with the geometric spatial density of the capillary network itself, via such underlying fields [15,42]. Unfortunately the theory of birth-and-growth processes, al- ready developed by the authors for volume growth [12], cannot be directly applied to analyze realistic models,, due to the peculiar- ities of the angiogenic process, and consequent mathematical difficulties. 0025-5564/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2008.03.009 * Corresponding author. Tel.: +39 0250316130; fax: +39 0250316172. E-mail addresses: Vincenzo.Capasso@unimi.it (V. Capasso), Alessandra. Micheletti@unimi.it (A. Micheletti), Daniela.Morale@unimi.it (D. Morale). Mathematical Biosciences 214 (2008) 20–31 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs