Tissue Development, Repair, and Remodeling - Tissue Functioning and Remodeling in the Circulatory and Ventilatory Systems - page 740

a growth-stress relationship in addition to the usual stress-strain relationship.
Bending experiments of excised vessel samples have been proposed to determine
the rheology constants of the wall layers during the layer structure-dependent
remodeling [ 1500 ]. However, in vitro data may be very far from in vivo values.

11.9 Growth Modeling

In planar cultures on matrices rich in collagen-4 and laminin, the endothelial cells
form clusters and pull on the matrix, generating tension lines that can extend
between the cell aggregates. The matrix eventually condenses along the tension
lines, along which the cells elongate and migrate, building cellular rods. The rate of
change in cell density is equal to the balance between the convection and the
strain-dependent motion. The inertia being negligible, the forces implicated in the
vasculogenesis model include the traction exerted by the cells on the extracellular
matrix, cell anchoring forces, and recoil forces of the matrix [ 1501 ]. A mathematical
model has been carried out using a finite difference scheme to study the role of
the mechanical and chemical forces in blood vessel formation, and to simulate
the formation of vascular networks in a plane [ 1502 ]. The numerical model
assumes: (1) traction forces exerted by the cells onto the extracellular matrix, (2) a
linear viscoelastic matrix, and (3) chemotaxis. The equation set is composed of:
(1) a convection-diffusion-reaction equation that describes the cell density, (2) a
conservation equation for extracellular matrix density, (3) a traction-displacement
equation associated with extracellular matrix organization, which contains a non-
linear term due to cell traction saturation at high cell densities, leading to two
scalar equations for the force and displacement vector components (2D problem),
and (4) diffusion-reaction equation for the chemotactic molecule concentration.
Spontaneous formation of networks can be explained via a purely mechanical
interaction between cells and the extracellular matrix.
Chemotaxis alone is not sufficient to generate tissue formation, with cell
proliferation and ECM formation. During vessel sprouting, mechanical forces can
help in the formation of well-defined vascular structures. The modeling can take
into account intercellular interactions and feeding [ 1503 ]. A model, which is based
on the Navier-Stokes equations in steady state and a simple mechanistic tissue
response, can predict bifurcation formation and microvessel separation in a porous
cellular medium [ 1504 ]. The tissue is remodeled according to the tangential shear
stress; the convection is approximated by simple non-diffusive heuristics at each
remodeling step.
Whereas angiogenesis cannot be explained by parabolic models, numerical
simulations based on hyperbolic models of chemotaxis mimic migration of en-
dothelial cells on Matrigel and the formation of networks that lead to vasculature
[ 1505 , 1506 ].