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The growth of tumor masses G. Dattoli ENEA FRASCATI

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

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  1. The growth of tumor massesG. DattoliENEA FRASCATI The point of view of a laser physicist (a theoritician)

  2. Power laws Math. Formulation • Self- Symilarity (Invariance under • Scale trasformation, Kallan-Szymanzik)

  3. TAYLOR-”Law”

  4. Bode-law • Distance of planets from the sun • n=n-th planet

  5. Astrophysics

  6. Biology& EchologyThe New fronteer • Volterra-Lotka, Malthus, Gompertz, Damouth, Kleiber…

  7. Echology: Damouth-law

  8. Biology & Ecology: the Paradise of the scaling law • Kleiber:mass- metabolyc rate

  9. Kleiber-Law 18-orders of magnitude!!!!!

  10. …3/4 ???

  11. Kleiber and dynamics… • Rate eq. (West, Brown, Enquist (1997))

  12. Eq. Di evoluzione

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

  14. Logistic-function,Gompertz…. • The solutions of the Eq. • Is a logistic type

  15. Analisi dei dati West & Brown)

  16. Growth of tumor masses Prostate cancer Mass (grams) of the human prostate cancer vs. time (days) using the WBE equation and the parameters

  17. Prostate and breast cancer andenergetic • age 40 years

  18. Tumor cell evolution Evolution of the tumor cell number vs time, final mass 671 g, different evolution times

  19. 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.

  20. 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

  21. Carrying Capacity and critical times for methastases spreading

  22. Tumor and methastases • Statistical model, Poissonian distribution • Il parametro is, along with the growth time, a measure of the tumour aggressivity

  23. Evolution of methastasis • 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

  24. Probability of 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)

  25. Angiogenesis

  26. Conclusions • 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

  27. …Conclusions • E=6 eV typical value of biochemical reactions

  28. Frattali e legge di Kleiber

  29. ….Fractal dimensions

  30. …

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