Numerical rate function determination in partial differential equations modeling cell population dynamics

Area of Science:

• Applied Mathematics
• Computational Biology
• Chemical Engineering

Background:

• Inverse problems in partial differential equations (PDEs) are often ill-posed, requiring regularization for stable solutions.
• Population balance equations (PBEs) model the evolution of cell populations based on size, crucial in biological and chemical systems.
• Accurate determination of rate functions within PDEs is essential for predictive modeling.

Purpose of the Study:

• To introduce a novel regularization method for solving inverse problems in size-structured population balance equations (PBEs).
• To determine an unknown rate function within a PBE using discrete measurements.
• To address the ill-posed nature of inverse problems by mitigating measurement error amplification.

Main Methods:

• The approximate inverse method, a pointwise regularization scheme, is employed.
• Separation of mollification in time and size variables is a key technique.
• Numerical instability is avoided by shifting differentiation to an analytically defined function.

Main Results:

• The developed method successfully determines unknown rate functions in size-structured PBEs.
• Numerical experiments with simulated data disturbances validate the scheme's performance.
• The regularization approach effectively handles measurement and model errors.

Conclusions:

• The proposed approximate inverse method offers a robust solution for inverse problems in PBEs.
• The technique provides a reliable way to estimate rate functions from discrete population data.
• This work contributes to accurate modeling of cell population dynamics through improved parameter estimation.