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 - PowerPoint PPT PresentationTRANSCRIPT
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
kk
k
x
xxy
xa
xkaxyxaxy
)()( xyxyxx k
TAYLOR-”Law”
5
12
)(
tE
tR
Bode-law
• Distance of planets from the sun
• n=n-th planet
1.0
)(
,73.1
,,44
),()(
amplitudewith
noffunctionyoscillatorweakn
b
unitsondependingbuta
nband n
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
MkR
)/(90
,
4
3
4
3
Kgscalk
MkR
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
3/4
const
MTRE
cL
,
Kleiber and dynamics…
• Rate eq. (West, Brown, Enquist (1997))
ratemetabolyctotalB
cellperEnergyE
rateMetabolycCellB
cellsofNumberN
td
NdEBNB
c
c
c
cccc
.
.
,
,
Eq. Di evoluzione
0
4
30
4
3
0
0
,
)(
,)(
,
mm
mE
Bm
E
mB
td
md
mBmB
mE
BmB
E
m
td
md
mNm
td
NdEBNB
c
c
c
c
c
c
c
c
cc
cccc
Living body Evolution
Von-Bartalanffy- Quantitative laws in metabolism and growth-Quarterly review on Biology 32, 217-231 (1957).
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)(
.)(,
,0
,
0
4
30
FELOttavianiLPGD
stransitionphaseLandauGinzburgh
BiolyBartalanffVonmbmadt
md
mm
mE
Bm
E
mB
td
md
c
c
c
c
Logistic-function,Gompertz….
• The solutions of the Eq.
• Is a logistic type
0
4
30
0
,
mm
mE
Bm
E
mB
td
md
c
c
c
c
)1ln(4
,
4
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M
m
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ET
B
mBM
c
c
c
c
4
3
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04
mB
nET cc
c
c
tt
B
E
eeM
mMtm
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,1)(
1
4
4 0 11
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
E
tPE
t
m
MEE
s
e
s
t
m
MEdttmBE
tc
t
ct
tc
c
ctc
tc
c
c
c
c
c
c
ts
s
s
c
ct
t
1164
324
33
0
4
4
00
4
3
03
4
4
3
4
4
4
3
10
4
3
0
102/10753.1,104.3/1094.2
))((85.0][,,,,4
1
,)(
4
1
,
,13
)1(4')'(
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
33
4
3,
4
1
cc
c
cT
mBP
tE
PP
3
4
4
)(
4
1
c
c
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
…