Computational Models of Vascularization and Therapy in Tumor Growth - Mechanical and Chemical Signaling in Angiogenesis - page 237

[ 73 ] on the interaction between ECM and fibroblasts, a cell well known for its
strong pulling force. The essential features of the models developed in this paper
and in the following ones [ 74 , 75 ] are the following:
• Cells exert traction forces onto the extra-cellular matrix, which is a viscoelastic
material;
• The Petri dish exerts a drag force on the matrix;
• Cells may move because of haptotaxis or chemotaxis (without any further
specification on the chemoattractant).
The model does not include any specific signaling process, but takes into
account mechanical issues. For this reason, it cannot describe phenomena which
depend on cell signaling, but it can describe the dependence on the formation of
the structure from the type of substratum.
It can be argued that a complete, realistic description of the diverse phases of in
vitro vasculogenesis should connect the migration regime described by the model
based on persistence and chemotaxis and the successive viscoelastic regime
described by the mechanical model or by some modification of it. This aim was
pursued by Tosin and coworkers [ 70 ]. They describe the system as made of two
layers, the gel and the ensemble of cells, obeying force balance equations
including a fundamental interaction term, which couples the motion of the cells
and the substratum. Their final conjecture is that migration and traction are two
different programs influenced by the local cell density, which lead toward a
behavior that is more ameboid-like or mesenchymal-like according to the envi-
ronmental conditions. In this view, persistence, chemotaxis, and mechanical
traction are complementary effects. The former two are essential in the early
migration-dominated phase of vasculogenesis, dictating the morphology of the
network and in particular the typical size of the polygons. The latter activates in a
later phase and after cell-to-cell contact and functions to stabilize the structure.
6 Concluding Remarks
In this chapter, we have briefly described different mathematical models related to
tumor growth and angiogenesis. Starting from the historical and first attempt to
describe tumor growth in vitro by using the Gompertz equation [ 4 - 7 ], we have
followed the development of those models that are now about to impact the
evaluation of oncological treatment efficacy in clinical trials [ 11 - 13 , 25 , 35 ].
Nowadays, attempts to complexify those models in order to take into account key
biological process such as angiogenesis is under development. This will require
the development of multiscale models integrating molecular signaling pathways
together with tissue regulation and morphological information on tumor growth
and regression when treated. Important methodological issues will have to be
addressed such as the multi-integration of data coming from different sources
(a priori knowledge from literature and data-driven knowledge from ad-hoc