Optimisation of Cancer Drug Treatments Using Cell Population Dynamics - Mathematical Methods and Models in Biomedicine - page 264

mechanistic way. It is usually done by adding a transport term in the left-hand side
of the reaction-diffusion equation

∂ p

t + ∇ . (

v p )= ∇ . ( D ∇ p )+ f ( p ) , (8)

∂

where f represents the reaction term and v the velocity of the transport movement.
The velocity v can be determined by using Darcy's law and, for instance, assump-
tions on the total amount of cells.
Authors generally consider that the population of tumour cells is submitted
to a growth signal representing all growth signals (inhibitors or promoters). The
equations governing the evolution of the density (or mass) of tumour cells and
of the concentration of chemicals are derived by applying the principle of mass
conservation to each species. The common form of such equations is

∂ p

t + ∇ . (

v p )= ∇ . ( D p ∇ p )+ α ( c , p ) − δ ( c , p ) , (9)

p p

∂

∂ c

t = ∇ . (

)+ α ( ) − δ ( , ) ,

D c ∇ c c c p (10)

c c

∂

where p is the density of tumour cells, c the concentration of the chemical (nutrients,
oxygen,

... ), D p the diffusion rate of tumour cells, α p represents their proliferation

rate, δ p their spontaneous death rate, D c is the diffusion rate of the chemical,

α c represents its production rate, δ c its degradation rate. Because tumour cell

proliferation and death depend on the concentration of the chemical, functions
α p and δ p depend, for instance linearly, on the concentration of the chemical

and on the tumour cell density. The function

α c depends, for instance linearly,
on the concentration of the chemical and the function δ c depends both on the

chemical concentration and on the density of tumour cells to model, for instance, the
consumption of the chemical substance by tumour cells. The same kind of equation
as Eq. ( 10 ) can be used to describe the evolution of the concentration of drug u .
As already mentioned in Sect. 2 , cancer therapies can have different effects
on tumour growth. To model the effect of a drug-inducing tumour cell death
(cytotoxic), one can add in the right-hand term of Eq. ( 9 ) a death term; thus the
equation governing the density of tumour cells submitted to the effect of a cytotoxic
drug may be given by

∂ p

t + ∇ . (

v p )= ∇ . ( D p ∇ p )+ α p ( c , p ) − δ p ( c , p ) − K cyto ( u , p ) , (11)

∂

where K cyto is a positive function that depends on the drug concentration and on the
tumour cell density, Eq. ( 10 ) remaining unchanged.
Instead, to model the effect on tumour growth of an anti-angiogenic therapy that
will reduce oxygen supply, one can add such a decay term in the right-hand term of
Eq. ( 10 ),whichthenbecomes

∂ c

t = ∇ . (

D c ∇ c )+ α c ( c ) − δ c ( c , p ) − K angio ( u , c ) , (12)

∂