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The growth of tumor masses G. Dattoli ENEA FRASCATI. The point of view of a laser physicist (a theoritician). Power laws. Math. Formulation Self- Symilarity (Invariance under Scale trasformation, Kallan-Szymanzik). TAYLOR-”Law”. Bode-law. Distance of planets from the sun

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The growth of tumor masses g dattoli enea frascati

The growth of tumor massesG. DattoliENEA FRASCATI

The point of view of a laser physicist

(a theoritician)

Power laws
Power laws

Math. Formulation

  • Self- Symilarity (Invariance under

  • Scale trasformation, Kallan-Szymanzik)

Bode law

  • Distance of planets from the sun

  • n=n-th planet

Biology echology the new fronteer
Biology& EchologyThe New fronteer

  • Volterra-Lotka, Malthus, Gompertz, Damouth, Kleiber…

Echology damouth law
Echology: Damouth-law

Kleiber law 18 orders of magnitude
Kleiber-Law 18-orders of magnitude!!!!!

3/4 ???

Kleiber and dynamics
Kleiber and dynamics…

  • Rate eq. (West, Brown, Enquist (1997))

Living body EvolutionVon-Bartalanffy- Quantitative laws in metabolism and growth-Quarterly review on Biology 32, 217-231 (1957).

Logistic function gompertz

  • The solutions of the Eq.

  • Is a logistic type

Growth of tumor masses
Growth of tumor masses

Prostate cancer

Mass (grams) of the human prostate cancer vs. time (days) using

the WBE equation and the parameters

Prostate and breast cancer and energetic
Prostate and breast cancer andenergetic

  • age 40 years

Tumor cell evolution
Tumor cell evolution

Evolution of the tumor cell number vs time, final mass

671 g, different evolution times

Tumor and host organ
Tumor and host organ

  • Human prostate cancer mass in grams (continuous line) and cancer metabolic rate in (continuous line), vs time in days (the dash curve refers to the average human metabolic rate). The cancer power density has been calculated assuming that the tumour has a spherical shape with a density comparable to that of the water.

Required power
Required Power

  • For a practically vanishing initial tumour mass and at small times we can evaluate the power associated to the tumour evolution, during its early stages is given by

  • while the energy used to generate the corresponding tumour mass is

Tumor and methastases
Tumor and methastases spreading

  • Statistical model, Poissonian distribution

  • Il parametro is, along with the growth time, a measure of the tumour aggressivity

Evolution of methastasis
Evolution of methastasis spreading

  • Probability vs. time (days) that s-malignant cells leaves the primary tumour

  • s=10 cells (solid line), s=50 cells (dash line), s=130 cells (dot line)

  • for M=671 g and

Probability of spreading
Probability of spreading spreading

  • Probability of colony formation vs. time (days) for a tumour with days and 1 colony (solid line), 10 colonies (dot line), 50 colonies (dash line), number of cells normalized to the saturation number (dash-dot), the parallel line corresponds to the clinical level (cells)

Angiogenesis spreading

Conclusions spreading

  • Biol. Evolution relies on complex mechanisms

  • Simple mathematical models are welcome

  • The same applies to tumor mass evolution

  • Concepts like carrying capacity e methastases spreading could be understoo in enegetic terms

  • The Kleiber “law” should be considered as the manifestation of a more general LAW

  • The dependence on the temperature should be included

…Conclusions spreading

  • E=6 eV typical value of biochemical reactions