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April  2022, 27(4): 1913-1926. doi: 10.3934/dcdsb.2021114

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 2022 Early access  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 and Continuous Dynamical Systems - B, 2022, 27 (4) : 1913-1926. doi: 10.3934/dcdsb.2021114
References:
[1]

T. Akamizu, K. Takaya, T. Irako, H. Hosoda, S. Teramukai, A. Matsuyama, H. Tada, K. Miura, A. Shimizu, M. Fukushima, et al., Pharmacokinetics, safety, and endocrine and appetite effects of ghrelin administration in young healthy subjects, European Journal of Endocrinology, 150 (2004), 447–455. doi: 10.1530/eje.0.1500447.

[2]

S. L. AronoffK. BerkowitzB. Shreiner and L. Want, Glucose metabolism and regulation: Beyond insulin and glucagon, Diabetes Spectrum, 17 (2004), 183-190.  doi: 10.2337/diaspect.17.3.183.

[3]

D. E. CummingsJ. Q. PurnellR. S. FrayoK. SchmidovaB. E. Wisse and D. S. Weigle, A preprandial rise in plasma ghrelin levels suggests a role in meal initiation in humans, Diabetes, 50 (2001), 1714-1719.  doi: 10.2337/diabetes.50.8.1714.

[4]

D. E. Cummings, Ghrelin and the short-and long-term regulation of appetite and body weight, Physiology & Behavior, 89 (2006), 71-84.  doi: 10.1016/j.physbeh.2006.05.022.

[5]

D. E. CummingsD. S. WeigleR. S. FrayoP. A. BreenM. K. MaE. P. Dellinger and J. Q. Purnell, Plasma ghrelin levels after diet-induced weight loss or gastric bypass surgery, New England Journal of Medicine, 346 (2002), 1623-1630.  doi: 10.1056/NEJMoa012908.

[6]

D. E. Cummings and M. H. Shannon, Roles for ghrelin in the regulation of appetite and body weight, Archives of Surgery, 138 (2003), 389-396. 

[7]

C. D. FryarQ. GuC. L. Ogden and K. M. Flegal, Anthropometric reference data for children and adults; united states, 2011-2014, Vital Health Stat, 3 (2016), 1-46. 

[8]

J. C. Hou and L. Min, J. E. Pessin and Insulin granule biogenesis, trafficking and exocytosis, Vitamins & Hormones, 80 (2009), 473-506. 

[9]

J. HuntJ. Smith and C. Jiang, Effect of meal volume and energy density on the gastric emptying of carbohydrates, Gastroenterology, 89 (1985), 1326-1330.  doi: 10.1016/0016-5085(85)90650-X.

[10]

M. Jacquier, F. Crauste, C. O. Soulage and H. A. Soula, A predictive model of the dynamics of body weight and food intake in rats submitted to caloric restrictions, PLoS One, 9 (2014).

[11]

G. L. Kellett and P. A. Helliwell, The diffusive component of intestinal glucose absorption is mediated by the glucose-induced recruitment of glut2 to the brush-border membrane, Biochemical Journal, 350 (2000), 155-162. 

[12]

G. L. Kellett, The facilitated component of intestinal glucose absorption, The Journal of Physiology, 531 (2001), 585-595.  doi: 10.1111/j.1469-7793.2001.0585h.x.

[13]

P. MaljaarsH. PetersD. Mela and A. Masclee, Ileal brake: A sensible food target for appetite control. a review, Physiology & Behavior, 95 (2008), 271-281.  doi: 10.1016/j.physbeh.2008.07.018.

[14]

B. K. Mani and J. M. Zigman, Ghrelin as a survival hormone, Trends in Endocrinology & Metabolism, 28 (2017), 843-854.  doi: 10.1016/j.tem.2017.10.001.

[15]

T. H. Moran and K. P. Kinzig, Gastrointestinal satiety signals ⅱ. cholecystokinin, American Journal of Physiology-Gastrointestinal and Liver Physiology, 286 (2004), G183–G188. doi: 10.1152/ajpgi.00434.2003.

[16]

J. MooreP. Christian and R. Coleman, Gastric emptying of varying meal weight and composition in man, Digestive Diseases and Sciences, 26 (1981), 16-22.  doi: 10.1007/BF01307971.

[17]

M. NakazatoN. MurakamiY. DateM. KojimaH. MatsuoK. Kangawa and S. Matsukura, A role for ghrelin in the central regulation of feeding, Nature, 409 (2001), 194-198.  doi: 10.1038/35051587.

[18]

D. L. Nelson, M. M. Cox and A. L. Lehninger, Principles of Biochemistry, Freeman New York, 2008.

[19]

J. OverduinR. S. FrayoH. J. GrillJ. M. Kaplan and D. E. Cummings, Role of the duodenum and macronutrient type in ghrelin regulation, Endocrinology, 146 (2005), 845-850.  doi: 10.1210/en.2004-0609.

[20]

P. PalumboS. DitlevsenA. Bertuzzi and A. De Gaetano, Mathematical modeling of the glucose–insulin system: A review, Mathematical Biosciences, 244 (2013), 69-81.  doi: 10.1016/j.mbs.2013.05.006.

[21]

P. PalumboS. Panunzi and A. De Gaetano, Qualitative behavior of a family of delay-differential models of the glucose-insulin system, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 399-424.  doi: 10.3934/dcdsb.2007.7.399.

[22]

S. Panunzi, P. Palumbo and A. De Gaetano, A discrete single delay model for the intra-venous glucose tolerance test,, Theoretical Biology and Medical Modelling, 4 (2007), 35. doi: 10.1186/1742-4682-4-35.

[23]

J. PiresA. BorriA. De GaetanoC. Manes and P. Palumbo, A short-term dynamical model for ghrelin, IFAC-PapersOnLine, 50 (2017), 11011-11016.  doi: 10.1016/j.ifacol.2017.08.2480.

[24]

J. G. Pires, Some insights into an integrative mathematical model: A prototype-model for bodyweight and energy homeostasis, Revista Eletrônica Gestão e Saúde, 3 (2016), 1271-1288. 

[25]

P. V. Röder, K. E. Geillinger, T. S. Zietek, B. Thorens, H. Koepsell and H. Daniel, The role of SGLT1 and GLUT2 in intestinal glucose transport and sensing, PloS One, 9 (2014).

[26]

D. E. Sadava, D. M. Hillis, H. C. Heller and M. Berenbaum, Life: The Science of Biology, Vol. 2, Macmillan, 2009.

[27]

S. Stanley, K. Wynne and S. Bloom, Gastrointestinal satiety signals iii. glucagon-like peptide 1, oxyntomodulin, peptide yy, and pancreatic polypeptide,, American Journal of Physiology-Gastrointestinal and Liver Physiology, 286 (2004), G693–G697. doi: 10.1152/ajpgi.00536.2003.

[28]

R. E. SteinertC. Feinle-BissetL. AsarianM. HorowitzC. Beglinger and N. Geary, Ghrelin, cck, glp-1, and pyy(3–36): Secretory controls and physiological roles in eating and glycemia in health, obesity and after rygb, Physiol Rev, 97 (2017), 411-463.  doi: 10.1152/physrev.00031.2014.

[29]

P. Toghaw, A. Matone, Y. Lenbury and A. De Gaetano, Bariatric surgery and t2dm improvement mechanisms: A mathematical model, Theoretical Biology and Medical Modelling, 9 (2012), 16.

[30]

C. Uluseker, G. Simoni, L. Marchetti, M. Dauriz, A. Matone and C. Priami, A closed-loop multi-level model of glucose homeostasis, PloS one, 13 (2018), e0190627. doi: 10.1371/journal.pone.0190627.

[31]

D. L. WilliamsD. E. CummingsH. J. Grill and J. M. Kaplan, Meal-related ghrelin suppression requires postgastric feedback, Endocrinology, 144 (2003), 2765-2767.  doi: 10.1210/en.2003-0381.

[32]

E. M. WrightM. G. Martìn and E. Turk, Intestinal absorption in health and disease-sugars, Best Practice & research Clinical Gastroenterology, 17 (2003), 943-956.  doi: 10.1016/S1521-6918(03)00107-0.

[33]

E. M. Wright, M. Sala-Rabanal, C. Ghezzi and D. D. Loo, Sugar absorption, in: Physiology of the Gastrointestinal Tract, Elsevier, 2018, 1051–1062.

[34]

X. YinY. LiG. XuW. An and W. Zhang, Ghrelin fluctuation, what determines its production?, Acta Biochimica et Biophysica Sinica, 41 (2009), 188-197.  doi: 10.1093/abbs/gmp001.

show all references

References:
[1]

T. Akamizu, K. Takaya, T. Irako, H. Hosoda, S. Teramukai, A. Matsuyama, H. Tada, K. Miura, A. Shimizu, M. Fukushima, et al., Pharmacokinetics, safety, and endocrine and appetite effects of ghrelin administration in young healthy subjects, European Journal of Endocrinology, 150 (2004), 447–455. doi: 10.1530/eje.0.1500447.

[2]

S. L. AronoffK. BerkowitzB. Shreiner and L. Want, Glucose metabolism and regulation: Beyond insulin and glucagon, Diabetes Spectrum, 17 (2004), 183-190.  doi: 10.2337/diaspect.17.3.183.

[3]

D. E. CummingsJ. Q. PurnellR. S. FrayoK. SchmidovaB. E. Wisse and D. S. Weigle, A preprandial rise in plasma ghrelin levels suggests a role in meal initiation in humans, Diabetes, 50 (2001), 1714-1719.  doi: 10.2337/diabetes.50.8.1714.

[4]

D. E. Cummings, Ghrelin and the short-and long-term regulation of appetite and body weight, Physiology & Behavior, 89 (2006), 71-84.  doi: 10.1016/j.physbeh.2006.05.022.

[5]

D. E. CummingsD. S. WeigleR. S. FrayoP. A. BreenM. K. MaE. P. Dellinger and J. Q. Purnell, Plasma ghrelin levels after diet-induced weight loss or gastric bypass surgery, New England Journal of Medicine, 346 (2002), 1623-1630.  doi: 10.1056/NEJMoa012908.

[6]

D. E. Cummings and M. H. Shannon, Roles for ghrelin in the regulation of appetite and body weight, Archives of Surgery, 138 (2003), 389-396. 

[7]

C. D. FryarQ. GuC. L. Ogden and K. M. Flegal, Anthropometric reference data for children and adults; united states, 2011-2014, Vital Health Stat, 3 (2016), 1-46. 

[8]

J. C. Hou and L. Min, J. E. Pessin and Insulin granule biogenesis, trafficking and exocytosis, Vitamins & Hormones, 80 (2009), 473-506. 

[9]

J. HuntJ. Smith and C. Jiang, Effect of meal volume and energy density on the gastric emptying of carbohydrates, Gastroenterology, 89 (1985), 1326-1330.  doi: 10.1016/0016-5085(85)90650-X.

[10]

M. Jacquier, F. Crauste, C. O. Soulage and H. A. Soula, A predictive model of the dynamics of body weight and food intake in rats submitted to caloric restrictions, PLoS One, 9 (2014).

[11]

G. L. Kellett and P. A. Helliwell, The diffusive component of intestinal glucose absorption is mediated by the glucose-induced recruitment of glut2 to the brush-border membrane, Biochemical Journal, 350 (2000), 155-162. 

[12]

G. L. Kellett, The facilitated component of intestinal glucose absorption, The Journal of Physiology, 531 (2001), 585-595.  doi: 10.1111/j.1469-7793.2001.0585h.x.

[13]

P. MaljaarsH. PetersD. Mela and A. Masclee, Ileal brake: A sensible food target for appetite control. a review, Physiology & Behavior, 95 (2008), 271-281.  doi: 10.1016/j.physbeh.2008.07.018.

[14]

B. K. Mani and J. M. Zigman, Ghrelin as a survival hormone, Trends in Endocrinology & Metabolism, 28 (2017), 843-854.  doi: 10.1016/j.tem.2017.10.001.

[15]

T. H. Moran and K. P. Kinzig, Gastrointestinal satiety signals ⅱ. cholecystokinin, American Journal of Physiology-Gastrointestinal and Liver Physiology, 286 (2004), G183–G188. doi: 10.1152/ajpgi.00434.2003.

[16]

J. MooreP. Christian and R. Coleman, Gastric emptying of varying meal weight and composition in man, Digestive Diseases and Sciences, 26 (1981), 16-22.  doi: 10.1007/BF01307971.

[17]

M. NakazatoN. MurakamiY. DateM. KojimaH. MatsuoK. Kangawa and S. Matsukura, A role for ghrelin in the central regulation of feeding, Nature, 409 (2001), 194-198.  doi: 10.1038/35051587.

[18]

D. L. Nelson, M. M. Cox and A. L. Lehninger, Principles of Biochemistry, Freeman New York, 2008.

[19]

J. OverduinR. S. FrayoH. J. GrillJ. M. Kaplan and D. E. Cummings, Role of the duodenum and macronutrient type in ghrelin regulation, Endocrinology, 146 (2005), 845-850.  doi: 10.1210/en.2004-0609.

[20]

P. PalumboS. DitlevsenA. Bertuzzi and A. De Gaetano, Mathematical modeling of the glucose–insulin system: A review, Mathematical Biosciences, 244 (2013), 69-81.  doi: 10.1016/j.mbs.2013.05.006.

[21]

P. PalumboS. Panunzi and A. De Gaetano, Qualitative behavior of a family of delay-differential models of the glucose-insulin system, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 399-424.  doi: 10.3934/dcdsb.2007.7.399.

[22]

S. Panunzi, P. Palumbo and A. De Gaetano, A discrete single delay model for the intra-venous glucose tolerance test,, Theoretical Biology and Medical Modelling, 4 (2007), 35. doi: 10.1186/1742-4682-4-35.

[23]

J. PiresA. BorriA. De GaetanoC. Manes and P. Palumbo, A short-term dynamical model for ghrelin, IFAC-PapersOnLine, 50 (2017), 11011-11016.  doi: 10.1016/j.ifacol.2017.08.2480.

[24]

J. G. Pires, Some insights into an integrative mathematical model: A prototype-model for bodyweight and energy homeostasis, Revista Eletrônica Gestão e Saúde, 3 (2016), 1271-1288. 

[25]

P. V. Röder, K. E. Geillinger, T. S. Zietek, B. Thorens, H. Koepsell and H. Daniel, The role of SGLT1 and GLUT2 in intestinal glucose transport and sensing, PloS One, 9 (2014).

[26]

D. E. Sadava, D. M. Hillis, H. C. Heller and M. Berenbaum, Life: The Science of Biology, Vol. 2, Macmillan, 2009.

[27]

S. Stanley, K. Wynne and S. Bloom, Gastrointestinal satiety signals iii. glucagon-like peptide 1, oxyntomodulin, peptide yy, and pancreatic polypeptide,, American Journal of Physiology-Gastrointestinal and Liver Physiology, 286 (2004), G693–G697. doi: 10.1152/ajpgi.00536.2003.

[28]

R. E. SteinertC. Feinle-BissetL. AsarianM. HorowitzC. Beglinger and N. Geary, Ghrelin, cck, glp-1, and pyy(3–36): Secretory controls and physiological roles in eating and glycemia in health, obesity and after rygb, Physiol Rev, 97 (2017), 411-463.  doi: 10.1152/physrev.00031.2014.

[29]

P. Toghaw, A. Matone, Y. Lenbury and A. De Gaetano, Bariatric surgery and t2dm improvement mechanisms: A mathematical model, Theoretical Biology and Medical Modelling, 9 (2012), 16.

[30]

C. Uluseker, G. Simoni, L. Marchetti, M. Dauriz, A. Matone and C. Priami, A closed-loop multi-level model of glucose homeostasis, PloS one, 13 (2018), e0190627. doi: 10.1371/journal.pone.0190627.

[31]

D. L. WilliamsD. E. CummingsH. J. Grill and J. M. Kaplan, Meal-related ghrelin suppression requires postgastric feedback, Endocrinology, 144 (2003), 2765-2767.  doi: 10.1210/en.2003-0381.

[32]

E. M. WrightM. G. Martìn and E. Turk, Intestinal absorption in health and disease-sugars, Best Practice & research Clinical Gastroenterology, 17 (2003), 943-956.  doi: 10.1016/S1521-6918(03)00107-0.

[33]

E. M. Wright, M. Sala-Rabanal, C. Ghezzi and D. D. Loo, Sugar absorption, in: Physiology of the Gastrointestinal Tract, Elsevier, 2018, 1051–1062.

[34]

X. YinY. LiG. XuW. An and W. Zhang, Ghrelin fluctuation, what determines its production?, Acta Biochimica et Biophysica Sinica, 41 (2009), 188-197.  doi: 10.1093/abbs/gmp001.

Figure 1.  Graphical scheme of the model. Continuous lines represent transfer of mass, dashed lines represent signals
Figure 2.  Plasma insulin and ghrelin evolutions
Figure 3.  Food volume in the gastrointestinal tract dynamics
Figure 4.  Plasma glucose dynamics
Figure 5.  Ghrelin dynamics in a day with 3 and with 2 meals
Table 1.  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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