Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 302

is a decreasing function of p . Indeed, due to limited amounts of nutrients, the
proliferation rate is a decreasing function of the population size whereas the rate
M is an increasing function and, typically, the net effect will be positive. We shall

use mainly the net proliferation rate since generally it is difficult to infer

( p )

Π and M

separately from external growth measurements. If Π ( 0 ) > M ( 0 ) ,thenitiseasyto

show that for all p ( 0 ) > 0 it follows that lim t → + ∞ p ( t )= K ,where K is the unique

solution of the equation . This value K , called the carrying capacity ,

then represents the maximum sustainable size of the population. Unfortunately, in
the vast majority of cases, the value K well exceeds values compatible with the life
of the host.
Initially, for a small tumor size p

Π ( p )= M ( p )

( 0 ) ,an exponential growth law is appropriate for

tumor growth. For p K higher-order terms can be neglected and approximatively

it holds that

p R ( p ( 0 )) p . (2)

However, as the tumor grows, these neglected terms matter. One of the most
commonly used laws to describe tumor growth is the Gompertz law [ 68 ],

R ( p )= a − b ln ( p ) , a > b > 0 . (3)

The parameter a represents a baseline proliferation rate, while b summarizes
the effects of mutual inhibitions between cells and competition for nutrients; it
sometimes is called the growth retardation factor. Normalizing p ( 0 )= 1, the tumor

size then becomes

exp a

e − bt p ( t )=

b (

1 − ) , (4)

which has a typical double exponential structure. The normalized carrying capacity
is K

exp b and thus it is convenient to rewrite R

= ( p ) in the form R ( p )=

ξ ln ( K / p ) , with the coefficient ξ a growth parameter that determines the rate of

convergence of p to K .
The Gompertz law belongs to the class of phenomenological growth models that
are based on competition between processes associated with proliferation and death.
The number of such models is amazingly large, and, as another prolific example, we
only mention the ubiquitous generalized logistic law,

a 1

p
K

ν

R ( p )= − , a > 0 , ν > 0 , (5)

notwithstanding the existence of various other models (e.g., see, [ 15 , 17 , 41 ]).
Indeed, both the Gompertz and logistic models were generalized in many ways,
and the recurrent question which model is more realistic [ 33 ] has no correct answer.
Since populations of cancer cells of different types and/or in different conditions
may behave very differently, it should not be surprising that models of cancer growth
can be so diversified. Indeed, any macroscopic growth law has to mirror a set of
phenomena that occur at the cellular scale including metabolic processes and inter-
cellular interactions that vary considerably from case to case [ 35 , 41 ]. Concerning
specifically Gompertz-like models, as pointed out in [ 39 , 68 ], all growth laws that