The growth of tumor massesG. Dattoli

ENEA FRASCATIThe point of view of a laser physicist

(a theoritician)

Power laws

Math. Formulation

• Self- Symilarity (Invariance under • Scale trasformation, Kallan-Szymanzik)

,)(1

),log()log())(log()(

00

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Astrophysics1.2rM

Biology& EchologyThe New fronteer

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

Echology: Damouth-law

25.2lP

Biology & Ecology: the Paradise of the scaling law

• Kleiber:mass- metabolyc rate

4

3

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)/(90

,

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

…3/4 ???

Card.rate -1/4

Card. period 1/4

Life Duration

1/4

Diam, Aorta 3/4

Mass. Brain 3/4

Consumption O(ml/s)

3/4

Gluc

mg/m

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const

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Kleiber and dynamics…

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

ratemetabolyctotalB

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Eq. Di evoluzione

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Living body Evolution

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

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• The solutions of the Eq.

• Is a logistic type

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Analisi dei dati West & Brown)

Growth of tumor masses

Prostate cancer

0 100 200 300

200

400

600

800

Mass (grams) of the human prostate cancer vs. time (days) using the WBE equation and the parameters

./1094.2)0(

,/10753.1)0(,103,101.2

4

33

0

695

dayJgB

dayJBgmJE

t

tccc

Prostate and breast cancer andenergetic

• age 40 years

0 50 100 150 200

200

400

600

800

0 2 4 6 8 101

10

100

1 103

1 104

1 105

1 106

WdayJBgWdaygB

daytJEM

mBB

m

aEBmBP

E

tPE

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tPE

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s

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s

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Tumor cell evolution

33

ccmnr

0.01 0.1 1 10 100 1 103

1 104

1

10

100

1 103

1 104

1 105

1 106

1 107

1 108

1 109

1 1010

1 1011

1 1012

n t 1 1( )

n t 4 1( )

n t 12 1( )

107

109

1011

t

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

days121

days481

days1441

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.

0 20 40 60 80 1000

200

400

600

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

•

4

3

0 )()( tmBtP

4

3

0

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4

3,

4

1

cc

c

cT

mBP

tE

PP

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)(

4

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c

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E

tPE

Carrying Capacity and critical times for methastases spreading

O

T

P

PC 3

4*

c

O

c

c

P

P

P

Et

3

4

0

*

B

Pm O

0.1 1 10 100 1 1031 10

3

0.01

0.1

1

10

100

1 103

1 104

1 105

1 106

P t( )

86400 2

10

86400

10

t

Tumor and methastases

• Statistical model, Poissonian distribution

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

))(exp(

!

)()( tn

s

tntp

s

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

days120,10 13

1 10 100 1 103

1 104

1 109

1 108

1 107

1 106

1 105

1 104

1 103

0.01

0.1

1

10

1 10 100 1 103

1 104

1 105

1 109

1 108

1 107

1 106

1 105

1 104

1 103

0.01

0.1

1

10

years5.6,10 16

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)

1 10 100 1 103

1 104

1 104

1 103

0.01

0.1

1

10

Angiogenesis

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

• Tk

Ei

eMTMB

4

3

),(

…Conclusions

• E=6 eV typical value of biochemical reactions

TK

Ei

eMcl

4

1

1

Frattali e legge di Kleiber

….Fractal dimensions

…