Abstract
Abstract We develop a computational method for simulating the nonlinear dynamics of an elastic tumor-host interface. This work is motivated by the recent linear stability analysis of a two-phase tumor model with an elastic membrane interface in 2D [47]. Unlike the classic tumor model with surface tension, the elastic interface condition is numerically challenging due to the 4th order derivative from the Helfrich bending energy. Here we are interested in exploring the nonlinear interface dynamics in a sharp interface framework. We consider a curvature dependent bending rigidity (curvature weakening [22]) to investigate metastasis patterns such as chains or fingers that invade the host environment. We solve the nutrient field and the Stokes flow field using a spectrally accurate boundary integral method, and update the interface using a nonstiff semi-implicit approach. Numerical results suggest curvature weakening promotes the development of branching patterns instead of encapsulated morphologies in a long period of time. For non-weakened bending rigidity, we are able to find self-similar shrinking morphologies based on marginally stable value of the apoptosis rate.
Highlights
A malignant tumor usually develops in a sequence of increasingly aggressive stages: carcinogenesis, avascular growth, angiogenesis and vascular growth [35]
The correctness of implementations of boundary integral methods for both Stokes ow and the modi ed Helmholtz equation was checked in a number of ways
We performed nonlinear simulations of a 2D, non-circular tumor with isotropic or curvature weakened bending rigidity growing in a host tissue
Summary
A malignant tumor usually develops in a sequence of increasingly aggressive stages: carcinogenesis, avascular growth, angiogenesis and vascular growth [35]. We derived a modi ed Laplace-Young condition of the stress jump across the interface for the Stokes equation using an energy variation approach, and performed a linear stability analysis to show how physical parameters such as viscosity, bending rigidity and apoptosis contribute to the morphological instability.
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