# Characterisation of Optimal Responses to Pulse Inputs in the Bergman Minimal Model

## TL;DR

This paper analyzes the Bergman minimal model to determine optimal insulin and carbohydrate input strategies that minimize plasma glucose peaks while maintaining a minimum concentration, providing necessary and sufficient conditions for optimal responses.

## Contribution

It introduces a method to specify insulin bolus timing and magnitude to minimize glucose peaks, and derives conditions for optimal plasma glucose regulation in the model.

## Key findings

- Optimal insulin bolus timing and magnitude identified.
- Necessary and sufficient conditions for glucose minimization established.
- Unavoidable peaks in plasma glucose are characterized.

## Abstract

The Bergman minimal model is a dynamic model of plasma glucose concentration. It has two input variables -- insulin delivery and carbohydrate intake. We investigate the behaviour of plasma glucose concentration predicted by the model given carbohydrate (CHO) inputs and commensurate insulin inputs. We observe that to maintain plasma glucose above a specified minimum concentration results in an unavoidable peak in plasma glucose. Additionally, we specify the timing and magnitude of a bolus pulse to minimise this unavoidable peak in plasma glucose concentration whilst attaining but not going below the desired minimum glucose concentration. Finally, we obtain necessary and sufficient conditions for the glucose concentration to be minimised.