Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 303

p p that tends to

produce a relative growth rate as the size p tends to zero are clearly

not adequate to describe the growth of small aggregate tumors whose doubling time,
a quantity related to a complex set of biological processes such as cell division cycle
and apoptosis, cannot be arbitrarily small. The Gomp-ex law by Wheldon [ 68 ]

∞

p

( a − b ln ( C )) if 0 < p < C ,

p = (6)

p ( a − b ln ( p )) if p ≥ C

provides a modification that uses the Gompertz law above a certain threshold C ,
but uses a simple exponential growth for smaller tumor sizes. Even if the tumor
size is measured in terms of cells, this stabilizes the ratio

p p

for small populations.

For another generalization of the Gompertz law, see [ 49 ].
All of these models were obtained by qualitative reasoning and then, for
specific cases, validated by means of data fitting, e.g., [ 36 , 37 ]. Some of them are
remarkably successful in the process of data-based validation. It is thus natural to
ask to what extent these models reproduce at a large degree of approximation finer
microscopical details, for example, of intercellular inhibitions. A second natural
question then arises as to whether these models can be unified in some general
framework. Can each of them be considered as a particular instance of some
metamodel? Among the few works aimed at introducing a mechanistic theory
that links macroscopic phenomenological models to microscopic interactions and
parameters, we cite the simple, yet plausible model in [ 35 ] which is based on the
realistic hypothesis of long range interactions between cells in a population whose
“structure is fractal.”This approach, significantly extended by one of us in [ 41 ],
allows to show that apparently contradictory growth models (logistic, generalized
logistic, Gompertzian, exponential, von Bertanlaffy, power law, del Santo-Guiot)
are simply macroscopic different manifestations of a common physical microscopic
framework . In other words, different values of the parameters of the microscopic law
result in different analytical laws for R . Thus, while one of these models may be

more appropriate depending on a specific medical situation, in principle they all
become viable alternatives in the investigation of the development of a tumor under
treatment.

( p )

3 Cancer Chemotherapy

Cancer chemotherapy, and notwithstanding significant and very interesting recent
developments towards novel cancer therapies such as immunotherapy, to this
day remains the elective non-surgical choice for treatment of tumors. Strictly
speaking, chemotherapy merely indicates the use of a chemical to cure a disease,
especially due to proliferating pathogens such as bacteria, tumor cells, etc. However,
chemotherapy has a so important clinical role in oncology that in the common
usage of language the word chemotherapy nowadays uniquely denotes anti-tumor