By B. D. Curti, D. L. Longo (auth.), John A. Adam, Nicola Bellomo (eds.)

Mathematical Modeling and Immunology an immense volume of human attempt and financial assets has been directed during this century to the struggle opposed to melanoma. the aim, in fact, has been to discover concepts to beat this difficult, demanding and likely never-ending fight. we will effectively think that even better efforts may be required within the subsequent century. The wish is that finally humanity could be winning; luck could have been completed while it truly is attainable to turn on and keep watch over the immune procedure in its festival opposed to neoplastic cells. facing the above-mentioned challenge calls for the fullest pos­ sible cooperation between scientists operating in several fields: biology, im­ munology, medication, physics and, we think, arithmetic. definitely, bi­ ologists and immunologists will make the best contribution to the re­ seek. besides the fact that, it really is now more and more well-known that arithmetic and desktop technological know-how may perhaps capable of make significant contributions to such prob­ lems. we can't count on mathematicians on my own to resolve basic prob­ lems in immunology and (in specific) melanoma learn, yet helpful sup­ port, although modest, might be supplied by means of mathematicians to the study aspirations of biologists and immunologists operating during this field.

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They are as follows: i) the tumor nodule is spherically symmetric at all times; rand t are the only independent variables; ii) cells die when the nutrient concentration a(r, t) falls below some critical level ai; iii) nutrient is consumed by living cells only; the consumption rate may depend on a(r, t) and on the cell proliferation rate; iv) necrotic cellular debris continually disintegrates into simpler chemical compounds that are freely permeable through the cell membranes. The cell volume lost in this way is replaced by cells pushed inward by the forces of adhesion and tension (see comments on this below); General Aspects of Modeling Tumor 35 v) the source of growth inhibition may be one of the following (or a combination of them): a) reduced nutrient concentration in the tumor interior.

31 ) with a corresponding equation for {3(r). In practice, the diffusion coefficients and sink/source terms have been chosen to have relatively simple forms for analytical convenience in the absence of any biological evidence to the contrary. 2; we have adopted in each case the notation of the authors concerned. One feature common to all these prevascular models is that the geometry is spherically symmetric: this is mathematically convenient, for in this ideal situation the net resultant of any intercellular forces within the colony is radially directed.

We might well seek to replace the word "physics" by the phrase "mathematical modeling" in the present context! 3 Other considerations As far as a mathematical model is concerned, the "world" can be divided into three components [BDa]: i) Things whose effects are neglected. ii) Things that affect the model but whose behavior the model is not designed to study. These are called independent (or exogenous variables). iii) Things the model is designed to study (the dependent or endogenous variables) .

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