Generalized sensitivity functions for size-structured population models

TL;DR

This paper extends generalized sensitivity functions to PDE-based size-structured population models, analyzing their application to the Smoluchowski coagulation equation to optimize experimental data collection.

Contribution

It introduces a novel PDE extension of GSFs for size-structured models and applies it to identify key data regions in coagulation processes.

Findings

GSFs can be effectively extended to PDE models.

Optimal data regions are identified for coagulation kernels.

Post-gelation data may be unnecessary for parameter estimation.

Abstract

Size-structured population models provide a popular means to mathematically describe phenomena such as bacterial aggregation, schooling fish, and planetesimal evolution. For parameter estimation, generalized sensitivity functions (GSFs) provide a tool that quantifies the impact of data from specific regions of the experimental domain. These functions help identify the most relevant data subdomains, which enhances the optimization of experimental design. To our knowledge, GSFs have not been used in the partial differential equation (PDE) realm, so we provide a novel PDE extension of the discrete and continuous ordinary differential equation (ODE) concepts of Thomaseth and Cobelli and Banks et al. respectively. We analyze the GSFs in the context of size-structured population models, and specifically analyze the Smoluchowski coagulation equation to determine the most relevant time and volume domains for three, distinct aggregation kernels. Finally, we provide evidence that parameter estimation for the Smoluchowski coagulation equation does not require post-gelation data.