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Rao F, Zhang Z, Li J. Dynamical analysis of a glucose-insulin regulatory system with insulin-degrading enzyme and multiple delays. J Math Biol 2023; 87:73. [PMID: 37856001 DOI: 10.1007/s00285-023-02003-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2021] [Revised: 12/28/2022] [Accepted: 09/18/2023] [Indexed: 10/20/2023]
Abstract
This paper investigates the dynamics of a glucose-insulin regulatory system model that incorporates: (1) insulin-degrading enzyme in the insulin equation; and (2) discrete time delays respectively in the insulin production term, hepatic glucose production term, and the insulin-degrading enzyme. We provide rigorous results of our model including the asymptotic stability of the equilibrium solution and the existence of Hopf bifurcation. We show that analytically and numerically at a certain value the time delays driven stability or instability occurs when the corresponding model has an interior equilibrium. Moreover, we illustrate the oscillatory regulation and insulin secretion via numerical simulations, which show that the model dynamics exhibit physiological observations and more information by allowing parameters to vary. Our results may provide useful biological insights into diabetes for the glucose-insulin regulatory system model.
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Affiliation(s)
- Feng Rao
- School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, 211816, Jiangsu, China.
| | - Zhongliang Zhang
- School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, 211816, Jiangsu, China
| | - Jiaxu Li
- Department of Mathematics, University of Louisville, Louisville, KY, 40292, USA.
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2
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Karamched B, Hripcsak G, Albers D, Ott W. Delay-induced uncertainty for a paradigmatic glucose-insulin model. CHAOS (WOODBURY, N.Y.) 2021; 31:023142. [PMID: 33653035 PMCID: PMC7910007 DOI: 10.1063/5.0027682] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2020] [Accepted: 02/02/2021] [Indexed: 06/12/2023]
Abstract
Medical practice in the intensive care unit is based on the assumption that physiological systems such as the human glucose-insulin system are predictable. We demonstrate that delay within the glucose-insulin system can induce sustained temporal chaos, rendering the system unpredictable. Specifically, we exhibit such chaos for the ultradian glucose-insulin model. This well-validated, finite-dimensional model represents feedback delay as a three-stage filter. Using the theory of rank one maps from smooth dynamical systems, we precisely explain the nature of the resulting delay-induced uncertainty (DIU). We develop a framework one may use to diagnose DIU in a general oscillatory dynamical system. For infinite-dimensional delay systems, no analog of the theory of rank one maps exists. Nevertheless, we show that the geometric principles encoded in our DIU framework apply to such systems by exhibiting sustained temporal chaos for a linear shear flow. Our results are potentially broadly applicable because delay is ubiquitous throughout mathematical physiology.
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Affiliation(s)
- Bhargav Karamched
- Department of Mathematics and Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, USA
| | - George Hripcsak
- Department of Biomedical Informatics, Columbia University, New York, New York 10032, USA
| | | | - William Ott
- Department of Mathematics, University of Houston, Houston, Texas 77204, USA
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3
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Chudtong M, Gaetano AD. A mathematical model of food intake. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2021; 18:1238-1279. [PMID: 33757185 DOI: 10.3934/mbe.2021067] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
The metabolic, hormonal and psychological determinants of the feeding behavior in humans are numerous and complex. A plausible model of the initiation, continuation and cessation of meals taking into account the most relevant such determinants would be very useful in simulating food intake over hours to days, thus providing input into existing models of nutrient absorption and metabolism. In the present work, a meal model is proposed, incorporating stomach distension, glycemic variations, ghrelin dynamics, cultural habits and influences on the initiation and continuation of meals, reflecting a combination of hedonic and appetite components. Given a set of parameter values (portraying a single subject), the timing and size of meals are stochastic. The model parameters are calibrated so as to reflect established medical knowledge on data of food intake from the National Health and Nutrition Examination Survey (NHANES) database during years 2015 and 2016.
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Affiliation(s)
- Mantana Chudtong
- Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
- Center of Excellence in Mathematics, the Commission on Higher Education, Si Ayutthaya Rd., Bangkok 10400, Thailand
| | - Andrea De Gaetano
- Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
- Consiglio Nazionale delle Ricerche, Istituto per la Ricerca e l'Innovazione Biomedica (CNR-IRIB), Palermo, Italy
- Consiglio Nazionale delle Ricerche, Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti" (CNR-IASI), Rome, Italy
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Das PN, Kumar A, Bairagi N, Chatterjee S. Effect of delay in transportation of extracellular glucose into cardiomyocytes under diabetic condition: a study through mathematical model. J Biol Phys 2020; 46:253-281. [PMID: 32583238 PMCID: PMC7441137 DOI: 10.1007/s10867-020-09551-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/11/2019] [Accepted: 05/26/2020] [Indexed: 01/02/2023] Open
Abstract
A four-dimensional model was built to mimic the cross-talk among plasma glucose, plasma insulin, intracellular glucose and cytoplasmic calcium of a cardiomyocyte. A time delay was considered to represent the time required for performing various cellular mechanisms between activation of insulin receptor and subsequent glucose entry from extracellular region into intracellular region of a cardiac cell. We analysed the delay-induced model and deciphered conditions for stability and bifurcation. Extensive numerical computations were performed to validate the analytical results and give further insights. Sensitivity study of the system parameters using LHS-PRCC method reveals that some rate parameters, which represent the input of plasma glucose, absorption of glucose by noncardiac cells and insulin production, are sensitive and may cause significant change in the system dynamics. It was observed that the time taken for transportation of extracellular glucose into the cell through GLUT4 plays an important role in maintaining physiological oscillations of the state variables. Parameter recalibration exercise showed that reduced input rate of glucose in the blood plasma or an alteration in transportation delay may be used for therapeutic targets in diabetic-like condition for maintaining normal cardiac function.
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Affiliation(s)
- Phonindra Nath Das
- Department of Mathematics, Memari College, Burdwan, West Bengal, 713146, India
| | - Ajay Kumar
- Non-communicable disease group, Translational Health Science and Technology Institute, NCR Biotech Science Cluster, Faridabad, 121001, India
| | - Nandadulal Bairagi
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata, 700032, India
| | - Samrat Chatterjee
- Complex Analysis Group, Translational Health Science and Technology Institute, NCR Biotech Science Cluster, Faridabad, 121001, India.
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Piemonte V, Capocelli M, De Santis L, Maurizi AR, Pozzilli P. A Novel Three-Compartmental Model for Artificial Pancreas: Development and Validation. Artif Organs 2017; 41:E326-E336. [DOI: 10.1111/aor.12980] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2017] [Revised: 04/09/2017] [Accepted: 05/17/2017] [Indexed: 12/12/2022]
Affiliation(s)
- Vincenzo Piemonte
- Faculty of Engineering, University Campus Biomedico of Rome; Rome Italy
| | - Mauro Capocelli
- Faculty of Engineering, University Campus Biomedico of Rome; Rome Italy
| | - Luca De Santis
- Faculty of Engineering, University Campus Biomedico of Rome; Rome Italy
| | - Anna Rita Maurizi
- Faculty of Engineering, University Campus Biomedico of Rome; Rome Italy
| | - Paolo Pozzilli
- Faculty of Engineering, University Campus Biomedico of Rome; Rome Italy
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Oviedo S, Vehí J, Calm R, Armengol J. A review of personalized blood glucose prediction strategies for T1DM patients. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2017; 33:e2833. [PMID: 27644067 DOI: 10.1002/cnm.2833] [Citation(s) in RCA: 107] [Impact Index Per Article: 15.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/15/2016] [Revised: 09/15/2016] [Accepted: 09/16/2016] [Indexed: 06/06/2023]
Abstract
This paper presents a methodological review of models for predicting blood glucose (BG) concentration, risks and BG events. The surveyed models are classified into three categories, and they are presented in summary tables containing the most relevant data regarding the experimental setup for fitting and testing each model as well as the input signals and the performance metrics. Each category exhibits trends that are presented and discussed. This document aims to be a compact guide to determine the modeling options that are currently being exploited for personalized BG prediction.
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Affiliation(s)
- Silvia Oviedo
- Institut d'Informàtica i Aplicacions, Parc Científic i Tecnològic de la Universitat de Girona, 17003, Girona, Spain
| | - Josep Vehí
- Institut d'Informàtica i Aplicacions, Universitat de Girona, Campus Montilivi, Edifici P4, 17071, Girona, Spain
| | - Remei Calm
- Institut d'Informàtica i Aplicacions, Universitat de Girona, Campus Montilivi, Edifici P4, 17071, Girona, Spain
| | - Joaquim Armengol
- Institut d'Informàtica i Aplicacions, Universitat de Girona, Campus Montilivi, Edifici P4, 17071, Girona, Spain
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Huard B, Bridgewater A, Angelova M. Mathematical investigation of diabetically impaired ultradian oscillations in the glucose-insulin regulation. J Theor Biol 2017; 418:66-76. [PMID: 28130099 DOI: 10.1016/j.jtbi.2017.01.039] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2016] [Revised: 01/16/2017] [Accepted: 01/22/2017] [Indexed: 11/17/2022]
Abstract
We study the effect of diabetic deficiencies on the production of an oscillatory ultradian regime using a deterministic nonlinear model which incorporates two physiological delays. It is shown that insulin resistance impairs the production of oscillations by dampening the ultradian cycles. Four strategies for restoring healthy regulation are explored. Through the introduction of an instantaneous glucose-dependent insulin response, explicit conditions for the existence of periodic solutions in the linearised model are formulated, significantly reducing the complexity of identifying an oscillatory regime. The model is thus shown to be suitable for representing the effect of diabetes on the oscillatory regulation and for investigating pathways to reinstating a physiological healthy regime.
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Affiliation(s)
- B Huard
- Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne NE1 8ST, UK.
| | - A Bridgewater
- Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne NE1 8ST, UK
| | - M Angelova
- Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne NE1 8ST, UK; School of Information Technology, Deakin University, Burwood Vic 3125, Australia
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Strilka RJ, Armen SB, Indeck MC. Qualitative analysis of subcutaneous Lispro and regular insulin injections for stress hyperglycemia: a pilot numerical study. J Theor Biol 2014; 356:192-200. [PMID: 24769252 DOI: 10.1016/j.jtbi.2014.04.023] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/23/2013] [Revised: 04/12/2014] [Accepted: 04/16/2014] [Indexed: 11/15/2022]
Abstract
Increased glucose variability (GV) is an independent risk factor for mortality in the critically ill; unfortunately, the optimal insulin therapy that minimizes GV is not known. We simulate the glucose-insulin feedback system to study how stress hyperglycemia (SH) states, taken to be a non-uniform group of physiologic disorders with varying insulin resistance (IR) and similar levels of hyperglycemia, respond to the type and dose of subcutaneous (SQ) insulin. Two groups of 100 virtual patients are studied: those receiving and those not receiving continuous enteral feeds. Stress hyperglycemia was facilitated by doubling the gluconeogenesis rate and IR was stepwise varied from a borderline to a high value. Lispro and regular insulin were simulated with dosages that ranged from 0 to 6 units; the resulting GV was analyzed after each insulin injection. The numerical model used consists of a set of non-linear differential equations with two time delays and five adjustable parameters. The results show that regular insulin decreased GV in both patient groups and rarely caused hypoglycemia. With continuous enteral feeds and borderline to mild IR, Lispro showed minimal effect on GV; however, rebound hyperglycemia that increased GV occurred when the IR was moderate to high. Without a nutritional source, Lispro worsened GV through frequent hypoglycemia episodes as the injection dose increased. The inferior performance of Lispro is a result of its rapid absorption profile; half of its duration of action is similar to the glucose ultradian period. Clinical trials are needed to examine whether these numerical results represent the glucose-insulin dynamics that occur in intensive care units, and if such dynamics are present, their clinical effects should be evaluated.
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Affiliation(s)
- Richard J Strilka
- Division of Trauma, Acute Care and Critical Care Surgery, Pennsylvania State College of Medicine, 500 University Drive, UPC II, Suite 3100, Hershey, PA 17033, United States.
| | - Scott B Armen
- Division of Trauma, Acute Care and Critical Care Surgery, Pennsylvania State College of Medicine, 500 University Drive, UPC II, Suite 3100, Hershey, PA 17033, United States
| | - Matthew C Indeck
- Division of Trauma, Acute Care and Critical Care Surgery, Pennsylvania State College of Medicine, 500 University Drive, UPC II, Suite 3100, Hershey, PA 17033, United States
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Abstract
In recent years it has been increasingly recognized that biochemical signals are not necessarily constant in time and that the temporal dynamics of a signal can be the information carrier. Moreover, it is now well established that the protein signaling network of living cells has a bow-tie structure and that components are often shared between different signaling pathways. Here we show by mathematical modeling that living cells can multiplex a constant and an oscillatory signal: they can transmit these two signals simultaneously through a common signaling pathway, and yet respond to them specifically and reliably. We find that information transmission is reduced not only by noise arising from the intrinsic stochasticity of biochemical reactions, but also by crosstalk between the different channels. Yet, under biologically relevant conditions more than 2 bits of information can be transmitted per channel, even when the two signals are transmitted simultaneously. These observations suggest that oscillatory signals are ideal for multiplexing signals.
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Affiliation(s)
- Wiet de Ronde
- FOM Institute AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands
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10
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Palumbo P, Ditlevsen S, Bertuzzi A, De Gaetano A. Mathematical modeling of the glucose–insulin system: A review. Math Biosci 2013; 244:69-81. [DOI: 10.1016/j.mbs.2013.05.006] [Citation(s) in RCA: 94] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2012] [Revised: 05/10/2013] [Accepted: 05/16/2013] [Indexed: 11/29/2022]
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Wu Z, Chui CK, Hong GS, Khoo E, Chang S. Glucose-insulin regulation model with subcutaneous insulin injection and evaluation using diabetic inpatients data. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2013; 111:347-356. [PMID: 23756090 DOI: 10.1016/j.cmpb.2013.05.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/13/2012] [Revised: 05/02/2013] [Accepted: 05/02/2013] [Indexed: 06/02/2023]
Abstract
Closed-loop insulin delivery systems often implement glucose measurement and insulin administration in the subcutis. However some existing models for glucose-insulin system ignored the dynamics of subcutaneous glucose and subcutaneously-injected insulin. This paper reports a two-compartment model that includes glucose and insulin dynamics in subcutis, and its evaluation using patient data. Clinical information such as glucose level, insulin dosage, insulin injection time and meals of anonymous diabetes inpatients was collected. Measured glucose level of the diabetic inpatients agrees with that of computer simulation. Due to the lack of glucose-insulin model with subcutaneously-injected insulin for type 2 diabetic patients, our model was compared with existing model for type 1 subjects. The new glucose-insulin model can mimic dynamics of glucose and insulin under the disturbance of insulin injections and meals. Model parameters were estimated using nonlinear least square method and their effect on pathology and physiology of diabetes were analyzed.
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Affiliation(s)
- Zimei Wu
- EA 04-06, Control and Mechatronics Lab 1, Department of Mechanical Engineering, 9 Engineering Drive 1, National University of Singapore, Singapore 117576, Singapore.
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13
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Wu X, Yang G, Zu Y, Zhou L. Molecular dynamics studies of β-hairpin folding with the presence of the sodium ion. Comput Biol Chem 2012; 38:1-9. [PMID: 22487489 DOI: 10.1016/j.compbiolchem.2012.02.003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/07/2011] [Revised: 01/20/2012] [Accepted: 02/14/2012] [Indexed: 11/19/2022]
Abstract
Metal ions are ubiquitous in protein systems and play a significant role during their folding processes. Nineteen independent structures were determined for the Na(+)/β-hairpin interacting systems, and their folding pathways are different. (i) For Na(S47), the turn is rapidly shaped with the help of Na(+) and acts as the folding nucleus for the rest regions. Two intermediate states are observed and the resulted structure is the most folded. (ii) For Na(B41), Na(B52), Na(B54), Na(S55) and Na(B56), the inclusive Na(+) ions are anchored by β-strands. The local structures around the Na(+) ions and the turn regions fold simultaneously and serve as two independent folding nuclei. (iii) The other systems have no folding nuclei and correspond to low-folded structures. Long-range electrostatic interactions contribute a lot to the folding, especially from the four negatively charged residues (Glu42, Asp46, Asp47 and Glu56). The initial positions of the Na(+) ions are largely responsible for the different folding behaviors. The interactions with sidechain- rather than backbone-O atoms generally lead to more compact structures. Another factor affecting the folding is whether the O atoms are associated with native H-bonds, and those involved show decreased affinities to metal ions. The addition of water solvent does not induce obvious folding and conformational transitions to the Na(+)/β-hairpin interacting systems.
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Affiliation(s)
- Xiaomin Wu
- Key Laboratory of Forest Plant Ecology, Ministry of Education, Northeast Forestry University, Harbin 150040, PR China
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