  Solid tumours have the ability to assemble their own vascular network for
optimizing their access to the vital nutrients. These new capillaries are
morphologically different from normal physiological vessels. In particular,
they have a much higher spatial tortuosity forcing an impaired flow within the
peritumoral area. This is a major obstacle for the efficient delivery of
antitumoral drugs. This work proposes a morpho-elastic model of the tumour
vessels. A tumour capillary is considered as a growing hyperelastic tube that
is spatially constrained by a linear elastic environment, representing the
interstitial matter. We assume that the capillary is an incompressible
neo-Hookean material, whose growth is modeled using a multiplicative
decomposition of the deformation gradient. We study the morphological stability
of the capillary by means of the method of incremental deformations superposed
on finite strains, solving the corresponding incremental problem using the
Stroh formulation and the impedance matrix method. The incompatible axial
growth of the straight capillary is found to control the onset of a bifurcation
towards a tortuous shape. The post-buckling morphology is studied using a mixed
finite element formulation in the fully nonlinear regime. The proposed model
highlights how the geometrical and the elastic properties of the capillary and
the surrounding medium concur to trigger the loss of marginal stability of the
straight capillary and the nonlinear development of its spatial tortuosity.
