The growth of tumor masses g dattoli enea frascati
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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

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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)


Taylor law

TAYLOR-”Law”


Bode law

Bode-law

  • Distance of planets from the sun

  • n=n-th planet


Astrophysics

Astrophysics


Biology echology the new fronteer

Biology& EchologyThe New fronteer

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


Echology damouth law

Echology: Damouth-law


Biology ecology the paradise of the scaling law

Biology & Ecology: the Paradise of the scaling law

  • Kleiber:mass- metabolyc rate


Kleiber law 18 orders of magnitude

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


The growth of tumor masses g dattoli enea frascati

…3/4 ???


Kleiber and dynamics

Kleiber and dynamics…

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


Eq di evoluzione

Eq. Di evoluzione


The growth of tumor masses g dattoli enea frascati

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


Logistic function gompertz

Logistic-function,Gompertz….

  • The solutions of the Eq.

  • Is a logistic type


Analisi dei dati west brown

Analisi dei dati West & Brown)


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


Carrying capacity and critical times for methastases spreading

Carrying Capacity and critical times for methastases spreading


Tumor and methastases

Tumor and methastases

  • Statistical model, Poissonian distribution

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


Evolution of methastasis

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


Probability of spreading

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)


Angiogenesis

Angiogenesis


Conclusions

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


Conclusions1

…Conclusions

  • E=6 eV typical value of biochemical reactions


Frattali e legge di kleiber

Frattali e legge di Kleiber


Fractal dimensions

….Fractal dimensions


The growth of tumor masses g dattoli enea frascati


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