In an extensive study of tumor growth Laird et al (1,2)  found that it traces the bi-parameter Gompertz function. Exp[a * (1 - Exp[ -b * t] ];  a and b are parameters and t is time.

Initially the tumor grows exponentially until crossing the inflection point at 2 whereupon its growth decelerates. As the tumor grows its core lacks oxygen and becomes necrotic. When the rate of growth equals the rate of death (necrosis) tumor size remains constant. Recently Gompertzian dynamics were reviewed by Waliszewski (3), and applied  by  Riffenburgh, and Johnstone to the analysis of cancer survival curves (4).

Cancer is viewed here as a metabolic deficiency and the present model illustrates the relationship between the metabolites A, B and the bi-modal hazard rate. The left graph depicts the change of the two metabolites with time.  A declines linearly. At a certain point it triggers the formation of a tumor that produces a substitute called here B

The graph on the right depicts the total metabolite level. The deficiency starts with a decline of A. At a certain point the low metabolite level triggers the formation of the tumor. Nevertheless the deficiency worsens, since the tumor is still too small to replenish fully the missing substance.  When the tumor catches up, the disease is compensated  and the patient feels healthy.  Gradually necrosis intensifies and   B production by the tumor  declines.

The next graph depicts the hazard rate =  (1 – normalized total metabolite level).

When the tumor is small the hazard rises (AB). Then it declines to the level when cancer was diagnosed (marked by the vertical line). At point C the deficiency  becomes severe and the hazard rate rises. The right  graph depicts hazard rates of regional breast cancer.

When lung cancer is diagnosed the disease is somewhat more advanced than in breast cancer  and the hazard rate declines .  The vertical line indicates when lung cancer was detected. The AB segment appears in cancers that are detected relatively early, e.g., breast, or uterus .and is missing in cancers that are detected later, e.g., pancreas.  


Only rarely does cancer treatment eliminate the entire tumor. If it does the hazard rate rises more than in an untreated disease . The tumor residues start growing and finally replenish the missing metabolite. Since their growth is postponed, the deficiency deepens and the hazard rises (right figure).

The model illustrates the close association between tumor growth and hazard rates.


1. Laird,  A.K. Laird, Dynamics of tumour growth,
Br. J. Cancer 18 (1964), pp. 490–502.
2. Laird, A.K. et al., , Dynamics of normal growth,
Growth 29 (1965), pp. 233–248.
3. Waliszewski  P
A principle of fractal-stochastic dualism and Gompertzian dynamics of growth and self-organization
Biosystems  82  (2005), pp. 61-73
Riffenburgh, RH, Johnstone, PAS  Survival Patterns of Cancer Patients
Cancer 91 (2001), pp.2469–75.

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