Mathematical model and its fast numerical method for the tumor growth.

Area of Science:

• Computational biology
• Mathematical modeling
• Biophysics

Background:

• The diffuse interface model is crucial for simulating tumor growth.
• The original model utilized the Cahn-Hilliard equation, which can be computationally intensive.
• Efficient and accurate modeling of tumor dynamics is essential for understanding cancer progression.

Purpose of the Study:

• To reformulate the diffuse interface model for tumor growth.
• To replace the fourth-order Cahn-Hilliard equation with a conservative second-order Allen-Cahn equation.
• To develop a computationally efficient and accurate model for tumor growth simulation.

Main Methods:

• The study employs a conservative second-order Allen-Cahn equation with a space-time dependent Lagrange multiplier.
• A recently developed hybrid numerical method is applied for numerical solutions.
• Various numerical experiments were conducted to validate the model.

Main Results:

• The new model demonstrates significant speed improvements compared to the original.
• The model effectively redistributes excess tumor mass from the interior to the boundary.
• Numerical results confirm the model's efficiency and accuracy.

Conclusions:

• The reformulated diffuse interface model offers a faster and effective approach to simulating tumor growth.
• The use of the Allen-Cahn equation provides computational advantages.
• The model's ability to manage mass distribution is a key feature for biological realism.