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### Open Access Journals

DCDS-B

We present a new, adaptive boundary integral method to simulate
solid tumor growth in 3-d. We use a reformulation of a classical
model that accounts for cell-proliferation, apoptosis, cell-to-cell
and cell-to-matrix adhesion. The 3-d method relies on accurate
discretizations of singular surface integrals, a spatial rescaling
and the use of an adaptive surface mesh. The discretized boundary
integral equations are solved iteratively using GMRES and a
discretized version of the Dirichlet-Neumann map, formulated in
terms of a vector potential, is used to determine the normal
velocity of the tumor surface. Explicit time stepping is used to
update the tumor surface. We present simulations of the nonlinear
evolution of growing tumors. At early times, good agreement is
obtained between the results of a linear stability analysis and
nonlinear simulations. At later times, linear theory is found to
overpredict the growth of perturbations. Nonlinearity results in
mode creation and interaction that leads to the formation of dimples
and the tumor surface buckles inwards. The morphologic instability
allows the tumor to increase its surface area, relative to its
volume, thereby allowing the cells in the tumor bulk greater access
to nutrient. This in turn allows the tumor to overcome the
diffusional limitations on growth and to grow to larger sizes than
would be possible if the tumor were spherical. Consequently,
instability provides a means for avascular tumor invasion.

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