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Cancer is the loss of the control mechanisms of cell proliferation: the balance between growth signals and inhibitory signals is broken, and cells acquire unlimited replication capacity. Physically, tumor growth can be described with mathematical models that evolve from pure exponential (initial phase, unrestricted) to sigmoid (established tumors, limited by nutrients, space and immunity).

The Gompertz model is the one that best fits the growth data of solid tumors: the growth rate decreases logarithmically as the tumor grows, reflecting the progressive limitation by hypoxia, nutrition and immune response. The inflection time t_inf marks the point of maximum growth rate (generally ~1/3 of the final size K_max).

Intratumoral hypoxia (pO < 5 mmHg) activates HIF-1, which induces angiogenesis (VEGF), anaerobic glycolysis (Warburg effect) and resistance to apoptosis. This mechanism transforms the tumor from a stationary mass limited by nutrients into an aggressive lesion with its own vascularization. Hypoxia also confers resistance to radiotherapy (O is radiosenitizing).

The accumulation of mutations follows a Poisson process with rate _mut 10/gene/division. Typically 510 driver mutations in key genes (TP53, RAS, BRCA, etc.) are needed to complete malignant transformation. The Iwata model for metastasis shows that the rate of dissemination scales with the tumor surface (N^(2/3)), explaining why large tumors have a higher metastatic risk.

Clinically, T_doubling time varies greatly: Burkitt lymphomas (2 days), breast cancer (40200 days), prostate cancer (300500 days). These values have direct implications for screening (window of opportunity for detection) and for chemotherapy planning (drugs that act in S phase are more effective in tumors with high GF).

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