Mathematical analysis of a tumour-immune interaction model: A moving boundary problem

Area of Science:

• Mathematical Biology
• Cancer Research
• Tumor Dynamics

Background:

• Cancer dormancy is a critical phase in tumor progression, characterized by a state of minimal or no growth.
• Understanding the mechanisms driving cancer dormancy is essential for developing effective therapeutic strategies.
• Immune surveillance plays a significant role in controlling tumor growth and potentially inducing dormancy.

Purpose of the Study:

• To develop and analyze a spatio-temporal mathematical model explaining cancer dormancy.
• To investigate the dynamics of tumor growth and invasion in both temporal and spatio-temporal contexts.
• To explore the role of cell motion, growth rates, and immune interactions in establishing and maintaining tumor dormancy.

Main Methods:

• Development of a moving boundary mathematical model for cancer dormancy.
• Stability analysis and numerical simulations of the temporal and spatio-temporal models.
• Application of the hyperbolic tangent method to determine travelling wave solutions.
• Calculation of minimum wave speeds for tumor cell invasion.

Main Results:

• The temporal model's stability analysis and simulations successfully replicate experimental observations of immune-induced tumor dormancy.
• Spatio-temporal analysis reveals that cell invasion dynamics are primarily governed by cell motion and growth rates.
• Numerical simulations demonstrate that tumors can swell to a dormant size.
• Stability analysis of the spatio-temporal model indicates the potential for dynamical stabilization of a tumor-free state.

Conclusions:

• The developed mathematical model provides insights into the mechanisms underlying cancer dormancy.
• Cellular motion and growth rates are key drivers of tumor invasion dynamics.
• The model supports the concept of immune-induced tumor dormancy and the possibility of achieving a stable tumor-free state through dynamical stabilization.