Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development.

*(English)*Zbl 0859.92012Summary: The growth and development of solid tumours occurs in two distinct stages – the avascular growth phase and the vascular growth phase. During the former growth phase the tumour remains in a diffusion-limited, dormant state of a few millimetres in diameter (cf. multicell spheroids, carcinoma in situ) while during the latter growth phase, invasion and metastasis may take place. In order to accomplish the transition from avascular to vascular growth, solid tumours may secrete diffusible substances known as tumour angiogenesis factors (TAF) into the surrounding tissue. Endothelial cells which form the lining of neighbouring blood vessels respond to this chemotactic stimulus in a well-ordered sequence of events. Capillary sprouts are formed which migrate towards the tumour, eventually penetrating it and permitting vascular growth to take place. This paper will present several mathematical models which deal with the various stages of growth and development of solid tumours.

##### MSC:

92C50 | Medical applications (general) |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |

##### Keywords:

diffusion equations; reaction-diffusion equations; development of solid tumours; avascular growth phase; vascular growth phase; tumour angiogenesis factors; chemotactic stimulus
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\textit{M. A. J. Chaplain}, Math. Comput. Modelling 23, No. 6, 47--87 (1996; Zbl 0859.92012)

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