A Multiscale Approach Leading to Hybrid Mathematical Models for Angiogenesis: The Role of Randomness
In biology and medicine we may observe a wide spectrum of formation of patterns, usually due to self-organization phenomena. This may happen at any scale; from the cellular scale of embryonic tissue formation, wound healing or tumor growth, and angiogenes
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1 Introduction In biology and medicine we may observe a wide spectrum of formation of patterns, usually due to self-organization phenomena. This may happen at any scale; from the cellular scale of embryonic tissue formation, wound healing or tumor growth, and angiogenesis to the much larger scale of animal grouping. Patterns are usually explained in terms of a collective behavior driven by “forces,” either external and/or internal, acting upon individuals (cells or organisms). In most of these organization phenomena, randomness plays a major role; here we wish to address the issue of the relevance of randomness as a key feature for producing nontrivial geometric patterns in biological structures. As working examples we offer a review of two important case studies involving angiogenesis, i.e., tumor-driven angiogenesis [7] and retina angiogenesis [8]. In both cases the reactants responsible for pattern formation are the cells organizing as a capillary network of vessels, and a family of underlying fields driving the organization, such as nutrients, growth factors, and alike [18, 19]. A fruitful approach to the mathematical description of such phenomena, suggested since long by various authors [16, 22, 26, 27, 30, 31], is based on the so-called individual based models, according to which the “movement” of each individual is described, embedded in the total population. This is also known as Lagrangian approach. Possible randomness is usually included in the motion, so that the variation in time of the (random) locations XNk (t) ∈ Rd , k = 1, . . . , N(t), of individuals in a group of size N(t) at time t ≥ 0 is described by a system of stochastic differential equations driven by gradients of suitable underlying fields. On the other hand the
V. Capasso () • D. Morale Department of Mathematics, University of Milan, 20133 Milan, Italy CIMAB (Interuniversity Centre for Mathematics Applied to Biology, Medicine and Environmental Sciences), University of Milan, 20133 Milan, Italy e-mail: [email protected]; [email protected] U. Ledzewicz et al., Mathematical Methods and Models in Biomedicine, Lecture Notes on Mathematical Modelling in the Life Sciences, DOI 10.1007/978-1-4614-4178-6 4, © Springer Science+Business Media New York 2013
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number N(t) of elements in the system may be subject to a stochastic dynamics, driven itself by the mentioned underlying fields. Vice versa these fields are usually strongly coupled with the dynamics of individuals in the population. The strong coupling of the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network, with the family of interacting underlying fields, is a major source of complexity from a mathematical and computational point of view. This is the reason why in literature we may find a large variety of mathematical models addressing some of the features of the angiogenic process, and still the problem of integration of all relevant features of the process is open [1, 9, 33, 34, 36, 40, 41]. Thus our
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