The growth of tumor masses
G. Dattoli
ENEA FRASCATI
The point of view of a laser physicist
(a theoritician)
Power laws
Math. Formulation
y ( x)  a x k  log( y ( x))  log( a )  k log( x),
 1
a  
 x0
k

 x
  y ( x)  

 x0
k

 ,

• Self- Symilarity (Invariance under
• Scale trasformation, Kallan-Szymanzik)
x  x  y ( x)  k y( x)
TAYLOR-”Law”
Et2 
R(t )  




1
5
Bode-law
•
•
Distance of planets from the sun
n=n-th planet
d ( n )  a bn ( n ),
a  44, but depending on units,
b  1.73,
 (n )  weak oscillator y function of n
with amplitude  0.1
Astrophysics
M r
2.1
Biology& Echology
The New fronteer
• Volterra-Lotka, Malthus, Gompertz,
Damouth, Kleiber…
Echology:
Damouth-law
P  l 2.25
Biology & Ecology: the Paradise of the scaling law
• Kleiber:massmetabolyc rate
RkM
3
4
3
4
RkM ,
3
4
k  90 cal /( s Kg )
Kleiber-Law 18-orders
of magnitude!!!!!
…3/4 ???
Card.rate
-1/4
Card.
1/4
period
Life
1/4
Duration
Diam, Aorta 3/4
Mass. Brain 3/4
Consumption
3/4
O(ml/s)
Gluc
mg/m
3/4
E  R T  M ,
 L c  const
Kleiber and dynamics…
• Rate eq. (West, Brown, Enquist (1997))
d Nc
B  N c Bc  Ec
,
dt
N c  Number of cells ,
Bc  Cell . Metabolyc rate
Ec  Energy per cell .
B  total metabolyc rate
Eq. Di evoluzione
B  N c Bc  Ec
d Nc
,
dt
m  N c mc
mc
Bc
d m


B ( m) 
m,
dt
Ec
Ec
B ( m )  B0 m
3
4
B0 mc
d m


m
dt
Ec
m0   m0
3
4
Bc

m,
Ec
Living body Evolution
Von-Bartalanffy- Quantitative laws in metabolism and growth-Quarterly review on Biology 32, 217-231 (1957).
d m B0mc 43 Bc

m  m,
dt
Ec
Ec
m0  m0 ,
dm


 a m  b m ,Von  Bartalanffy  ( Biol .)
dt
Ginzburgh  Landau ( phase  transitions )
D. G. , P. L. Ottaviani  FEL...
Logistic-function,
Gompertz….
• The solutions of the Eq.
3
d m B0 mc 4 Bc

m  m,
dt
Ec
Ec
m0  m0
• Is a logistic type
 m 
m(t )  M  4 0 e 1
 M
4 Ec
1 
Bc
t
 B0 mc
  M  
 B
 1  e 1  ,
c

4
t

T 4
4


 ,

T 
1
4
Ec
m 
ln( 1   0  )
Bc
 M 
4 Ec nc0
3
4
0
B0 m
Analisi dei dati West & Brown)
Growth of tumor masses
Prostate cancer
Mass (grams) of the human prostate cancer vs. time (days) using
the WBE equation and the parameters
Ec  2.110 5 J , mc  3 10 9 g , Bct (0)  1.753 10 6 J / day,
B (0)  2.94 10 g
t
0
3

3
4
J / day.
800
600
400
200
0
100
200
300
Prostate and breast cancer and
energetic
800
• age 40 years
600
400
1 10
6
1 10
5
1 10
4
1 10
3
100
10
200
1
0
50
100
150
0
2
200
4
t
s

3

3


ME
t
1

e
c 
E  B0t  m(t ' ) dt '  4
  (1) s  

m

s
s 1
s
c
0

t
3
4
6
8
10

,


4
MEc  t 
E
  ,
mc   
1 ( Pc t ) 4
E 4
,
4 Ec 3
3
aEc
B0t mc
1 Pc t 
t
t
t
4
,
P

B
m
,
B

,
B

, E[ J ]  0.85 (t ( day )) 4
c
0
c
0
c
4
3
4
4 Ec
mc
M
4
E
B  2.94  10 g
t
0
3

3
4
2
/ day  3.4 10 W  g

3
4
, Bct  1.753 10 6 J / day  2 10 11W
Tumor cell evolution
r3
3 nc mc
 
Evolution of the tumor cell number vs time, final mass
671 g, different evolution times
1 10
12
 1  48 days
1 10
11
1 10
10
 1  12 days
n( t  1  1) 1 109
n( t  4  1) 1 108
n( t  12  1) 1 10
7
7
 1  144 days
10
1 10
6
1 10
5
9
10
11
10
1 10
4
1 10
3
100
10
1
0.01
0.1
1
10
t
100
1 10
3
1 10
4
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.
600
400
200
0
0
20
40
60
80
100
Required Power
3
4
• P(t )  B0 m(t )
• 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
•
1 Pc4 3
PT  3
t ,
4 E c3
Pc  B0 m
3
4
c
• while the energy used to generate the
corresponding tumour mass is
•
1 ( Pc t ) 4
E 4
4 E c3
Carrying Capacity and critical times
for methastases spreading
P
C T
PO
t* 
4 Ec
Pc
3
PO
Pc
1 10
6
P 
m*  3  O 
 B0 
4
1 10
5
1 10
4
P( t )
1 10
86400 2
100
10
10
86400
1
10
3
0.1
0.01
1 10
3
0.1
1
10
t
100
1 10
3
Tumor and methastases
• Statistical model, Poissonian distribution

 n(t )s
p (t ) 
exp(  n(t ))
s!
• Il parametro  is, along with the growth time,
a measure of the tumour aggressivity
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
  106 , 1  6.5 years
  103 , 1  120 days
10
10
1
1
0.1
0.1
0.01
0.01
1 10
3
1 10
3
1 10
4
1 10
4
1 10
5
1 10
5
1 10
1 10
6
6
7
1 10
7
1 10
8
1 10
8
1 10
9
1 10
9
1 10
1
10
100
1 10
3
1 10
4
1
10
100
1 10
3
1 10
4
1 10
5
Probability of spreading
10
1
0.1
0.01
1 10
3
1 10
4
1
•
10
100
3
1 10
1 10
4
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
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
•
3  Ei
kT
4
B( M , T )  M e
…Conclusions
1
4
l  c1M e
•

Ei
KT
E=6 eV typical value of biochemical reactions
Frattali e legge di Kleiber
….Fractal dimensions
…