A model for the hepatic glucose metabolism based on Hill and step functions

We propose a system of first-order ordinary differential equations to describe and understand the physiological mechanisms of the interplay between plasma glucose and insulin and their behaviors in diabetes. The proposed model is based on Hill and step functions which are used to simulate the switch-like behavior that occurs in metabolic regulatory variables when some of the threshold parameters are approached. A simplified piecewise-linear system is also proposed to study the possible equilibria and solutions and used to introduce simple theoretical control mechanisms representing the action of an artificial pancreas and regulating exogenous insulin.

[1]  Ganesh Sriram,et al.  Mathematical Modeling: Bridging the Gap between Concept and Realization in Synthetic Biology , 2010, Journal of biomedicine & biotechnology.

[2]  Luca Dieci,et al.  A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side , 2012, J. Comput. Appl. Math..

[3]  Giancarlo Ferrari-Trecate,et al.  The Switching Threshold Reconstruction Problem for Piecewise-Affine Models of Genetic Regulatory Networks , 2008, IEEE Transactions on Automatic Control.

[4]  H. D. Jong,et al.  Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability , 2006, Journal of mathematical biology.

[5]  Luca Dieci,et al.  Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field , 2011, Numerische Mathematik.

[6]  H. D. Jong,et al.  Qualitative simulation of genetic regulatory networks using piecewise-linear models , 2004, Bulletin of mathematical biology.

[7]  Peter Butler,et al.  Pulsatile insulin secretion: detection, regulation, and role in diabetes. , 2002, Diabetes.

[8]  E. Mosekilde,et al.  Modeling the insulin-glucose feedback system: the significance of pulsatile insulin secretion. , 2000, Journal of theoretical biology.

[9]  Luca Dieci,et al.  A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis☆ , 2013 .

[10]  Nicoletta Del Buono,et al.  On the Equivalence between the Sigmoidal Approach and Utkin's Approach for Piecewise-Linear Models of Gene Regulatory Networks , 2014, SIAM J. Appl. Dyn. Syst..

[11]  G. Yagil,et al.  On the relation between effector concentration and the rate of induced enzyme synthesis. , 1971, Biophysical journal.

[12]  G. Calamita,et al.  Aquaporin Membrane Channels: Biophysics, Classification, Functions, and Possible Biotechnological Applications , 2011 .

[13]  Jean-Luc Gouzé,et al.  Exact control of genetic networks in a qualitative framework: The bistable switch example , 2011, Autom..

[14]  Hidde de Jong,et al.  Modeling and Simulation of Genetic Regulatory Systems: A Literature Review , 2002, J. Comput. Biol..

[15]  Y. Kuang,et al.  Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays. , 2006, Journal of theoretical biology.

[16]  Claudio Cobelli,et al.  Meal Simulation Model of the Glucose-Insulin System , 2007, IEEE Transactions on Biomedical Engineering.

[17]  G. Calamita,et al.  Insulin- and leptin-mediated control of aquaglyceroporins in human adipocytes and hepatocytes is mediated via the PI3K/Akt/mTOR signaling cascade. , 2011, The Journal of clinical endocrinology and metabolism.

[18]  Xinyu Song,et al.  Modeling Impulsive Injections of Insulin: Towards Artificial Pancreas , 2012, SIAM J. Appl. Math..

[19]  Nicoletta Del Buono,et al.  Event Driven Approach for Simulating Gene Regulation Networks , 2014, ICCSA.

[20]  R. Bergman.  Toward Physiological Understanding of Glucose Tolerance: Minimal-Model Approach , 1989, Diabetes.

[21]  Yang Kuang,et al.  Enhanced modelling of the glucose–insulin system and its applications in insulin therapies , 2009, Journal of biological dynamics.

[22]  G. Calamita,et al.  Biophysical assessment of aquaporin‐9 as principal facilitative pathway in mouse liver import of glucogenetic glycerol , 2012, Biology of the cell.

[23]  J. Gouzé,et al.  A class of piecewise linear differential equations arising in biological models , 2002 .

[24]  G. Calamita,et al.  Liver Glycerol Permeability and Aquaporin-9 Are Dysregulated in a Murine Model of Non-Alcoholic Fatty Liver Disease , 2013, PloS one.

[25]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[26]  E. Mosekilde,et al.  Computer model for mechanisms underlying ultradian oscillations of insulin and glucose. , 1991, The American journal of physiology.

[27]  James D. Murray.  Mathematical Biology: I. An Introduction , 2007 .

[28]  Yang Kuang,et al.  Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview , 2006 .

[29]  G. Calamita,et al.  Reduced hepatic aquaporin-9 and glycerol permeability are related to insulin resistance in non-alcoholic fatty liver disease , 2014, International Journal of Obesity.