# American Institute of Mathematical Sciences

## A short-term food intake model involving glucose, insulin and ghrelin

 1 High School Melchiorre Delfico, Teramo, Italy 2 CNR-IASI Biomathematics Laboratory, National Research Council of Italy, Rome, Italy 3 CNR-IRIB Institute for Biomedical Research and Innovation, National Research Council of Italy, Palermo, Italy 4 Department of Information Engineering, Computer Science, and Mathematics, University of L'Aquila, L'Aquila, Italy 5 Department of Biotechnologies and Biosciences, University of Milano-Bicocca, Milan, Italy 6 Centro de Desenvolvimento Tecnológico em Saúde/Oswaldo Cruz Foundation, Rio de Janeiro, Brazil

* Corresponding author: Alessandro Borri (e-mail: alessandro.borri@biomatematica.it)

Received  December 2020 Published  April 2021

Body weight control is gaining interest since its dysregulation eventually leads to obesity and metabolic disorders. An accurate mathematical description of the behavior of physiological variables in humans after food intake may help in understanding regulation mechanisms and in finding treatments. This work proposes a multi-compartment mathematical model of food intake that accounts for glucose-insulin homeostasis and ghrelin dynamics. The model involves both food volumes and glucose amounts in the two-compartment system describing the gastro-intestinal tract. Food volumes control ghrelin dynamics, whilst glucose amounts clearly impact on the glucose-insulin system. The qualitative behavior analysis shows that the model solutions are mathematically coherent, since they stay positive and provide a unique asymptotically stable equilibrium point. Ghrelin and insulin experimental data have been exploited to fit the model on a daily horizon. The goodness of fit and the physiologically meaningful time courses of all state variables validate the efficacy of the model to capture the main features of the glucose-insulin-ghrelin interplay.

Citation: Massimo Barnabei, Alessandro Borri, Andrea De Gaetano, Costanzo Manes, Pasquale Palumbo, Jorge Guerra Pires. A short-term food intake model involving glucose, insulin and ghrelin. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021114
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##### References:
Graphical scheme of the model. Continuous lines represent transfer of mass, dashed lines represent signals
Plasma insulin and ghrelin evolutions
Food volume in the gastrointestinal tract dynamics
Plasma glucose dynamics
Ghrelin dynamics in a day with 3 and with 2 meals
Model parameters and initial conditions
 Parameter Units Value Reference $r$ $ml/min$ $35$ [23] $k^{max}_{JS}$ $min^{-1}$ $0.0201$ Identification $\lambda_{JS}$ $min^{-1}$ $9.1871 \cdot 10^{-4}$ Identification $k_S,k_J$ $mml/min$ $6.2568$ Identification $k_{XJ}$ $min^{-1}$ $0.0737$ Identification $Ca$ $-$ $0.5558$ [3] $k_{GJ}$ $ml/min$ $50.1503$ Identification $k_G$ $mmol/min$ $0.2066$ Steady State $V_G$ $l$ $10.483$ [22] $BW$ $kg$ $68.97$ [3,7] $k_{xGI}$ $min^{-1}$ $5.3\cdot 10^{-5}$ [22] $k_{IRG}$ $min^{-1}$ $0.0049$ Steady state $\gamma_{IRG}$ $-$ $3.0763$ Identification $V_I=V_H$ $l$ $17.2425$ [22] $k_{xI}$ $min^{-1}$ $0.059$ [21] $k_{RG}$ $min^{-1}$ $17.6948$ Steady state $k^{min}_H$ $pmol/ml$ $650407.4627$ Identification $k^{max}_H$ $pmol/ml$ $899990.4238$ Identification $t_{H}$ $pmol/ml$ $1195.2917$ Identification $\lambda_{HJ}$ $min^{-1}$ $0.007$ Identification $k_{XH}$ $l$ $0.239$ [1] $S_0$ $ml$ $363.3046$ Steady state $J_0$ $ml$ $169.8402$ Steady state $G_{S0}$ $mmol$ $0$ Steady state $G_{J0}$ $mmol$ $0$ Steady state $G_0=G_b$ $mM$ $4.6239$ Identification $I_0$ $pM$ $80.4264$ [3] $R_0$ $pmol$ $16581.6656$ Identification $H_0$ $pg/ml$ $524.5618$ Identification
 Parameter Units Value Reference $r$ $ml/min$ $35$ [23] $k^{max}_{JS}$ $min^{-1}$ $0.0201$ Identification $\lambda_{JS}$ $min^{-1}$ $9.1871 \cdot 10^{-4}$ Identification $k_S,k_J$ $mml/min$ $6.2568$ Identification $k_{XJ}$ $min^{-1}$ $0.0737$ Identification $Ca$ $-$ $0.5558$ [3] $k_{GJ}$ $ml/min$ $50.1503$ Identification $k_G$ $mmol/min$ $0.2066$ Steady State $V_G$ $l$ $10.483$ [22] $BW$ $kg$ $68.97$ [3,7] $k_{xGI}$ $min^{-1}$ $5.3\cdot 10^{-5}$ [22] $k_{IRG}$ $min^{-1}$ $0.0049$ Steady state $\gamma_{IRG}$ $-$ $3.0763$ Identification $V_I=V_H$ $l$ $17.2425$ [22] $k_{xI}$ $min^{-1}$ $0.059$ [21] $k_{RG}$ $min^{-1}$ $17.6948$ Steady state $k^{min}_H$ $pmol/ml$ $650407.4627$ Identification $k^{max}_H$ $pmol/ml$ $899990.4238$ Identification $t_{H}$ $pmol/ml$ $1195.2917$ Identification $\lambda_{HJ}$ $min^{-1}$ $0.007$ Identification $k_{XH}$ $l$ $0.239$ [1] $S_0$ $ml$ $363.3046$ Steady state $J_0$ $ml$ $169.8402$ Steady state $G_{S0}$ $mmol$ $0$ Steady state $G_{J0}$ $mmol$ $0$ Steady state $G_0=G_b$ $mM$ $4.6239$ Identification $I_0$ $pM$ $80.4264$ [3] $R_0$ $pmol$ $16581.6656$ Identification $H_0$ $pg/ml$ $524.5618$ Identification
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