A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth.

Area of Science:

• Computational fluid dynamics
• Mathematical modeling
• Biophysics

Background:

• Interface evolution problems often involve complex boundary conditions.
• Accurate discretization of derivatives and jump conditions is crucial for numerical stability.
• Modeling biological processes like tumor growth requires handling heterogeneous environments.

Purpose of the Study:

• To develop a novel ghost cell/level set method for interface evolution governed by reaction-diffusion equations.
• To accurately handle curvature-dependent boundary conditions and derivative jumps.
• To simulate glioblastoma tumor growth in heterogeneous brain tissue.

Main Methods:

• Utilized a ghost cell method for accurate discretization of normal derivative jump boundary conditions.
• Developed a new iterative solver for linear and nonlinear quasi-steady reaction-diffusion equations.
• Employed adaptive discretization for curvature and normal vector computation.
• Introduced a novel discrete approximation for the Heaviside function.

Main Results:

• Achieved better than 1.5-order convergence in numerical examples, outperforming traditional methods.
• Successfully applied the method to model tumor growth with nonlinear nutrient and pressure equations.
• Simulated glioblastoma growth in a heterogeneous 1 cm brain tissue model.

Conclusions:

• The developed ghost cell/level set method offers superior accuracy and convergence for interface evolution problems.
• The simulation results highlight the significant impact of tissue heterogeneity on tumor growth morphology.
• This approach provides a robust tool for studying complex biological phenomena like cancer progression.