To use or not to use? Backward equations in stochastic carcinogenesis models

Area of Science:

• Mathematical biology
• Stochastic processes
• Cancer modeling

Background:

• Multistage carcinogenesis models are crucial for understanding cancer development.
• Little's method (1995) using Kolmogorov backward equations is a standard for hazard function computation.
• This method encounters limitations with time-dependent model parameters.

Purpose of the Study:

• To address the failure of Little's method for time-dependent parameters in carcinogenesis models.
• To introduce and discuss an alternative computational method based on the Kolmogorov forward equation.
• To elucidate the relationships between forward and backward Kolmogorov equations and the characteristic method.

Main Methods:

• The study analyzes the limitations of the Kolmogorov backward equation approach.
• It proposes and details an alternative method utilizing the Kolmogorov forward equation.
• The characteristic method is employed to explore the interplay between the two equations.

Main Results:

• The Kolmogorov backward equation method is shown to be inadequate for time-dependent parameters.
• The Kolmogorov forward equation provides a viable alternative for hazard function computation in such scenarios.
• The interconnectedness of the forward, backward, and characteristic methods is highlighted.

Conclusions:

• The Kolmogorov forward equation offers a robust alternative for hazard function calculation in time-dependent multistage carcinogenesis models.
• Understanding the interplay between different mathematical methods is essential for accurate modeling.
• The findings provide a valuable tool for researchers in cancer modeling and risk assessment.