This study concentrates on Type 1 diabetes Affects 4 20 per 100000 with peak

This study concentrates on type 1 diabetes affects 4

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This study concentrates on Type 1 diabetes Affects 4-20 per 100,000 with peak occurrence around 14 years of age Causes serious health conditions, especially heart disease and nerve damage Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (23/68)
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Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Glucose Metabolism Glucose Metabolism Ingest food for nutrients and energy Carbohydrates are broken into simple sugars Sugars are absorbed into the blood Cells access blood sugar for energy Glucose Control in Blood High glucose levels are bad for tissues (osmotic pressure?) β -cells in pancreas sense high levels and release insulin Insulin facilitates glucose entering tissues (skeletal muscle, esp.) Convert glucose to glycogen to store in liver Negative feedback control Many other controlling hormones Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (24/68)
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Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Modeling Glucose Metabolism General Glucose Control Model Let G ( t ) be the blood glucose level and I ( t ) be the blood insulin level A general differential equation describing this system is dG dt = f 1 ( G, I ) + J ( t ) , dI dt = f 2 ( G, I ) , where J ( t ) is the external uptake of glucose (a control function ) Many significantly more complex models exist The body wants to maintain homeostasis, so assume an equilibrium ( G 0 , I 0 ) or f 1 ( G 0 , I 0 ) = 0 and f 2 ( G 0 , I 0 ) = 0 . We examine the translated variables (about equilibrium) g ( t ) = G ( t ) - G 0 and i ( t ) = I ( t ) - I 0 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (25/68)
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Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Linearization 1 Taylor’s Theorem for Two Variables allows the expansion of the functions f 1 ( G, I ) and f 2 ( G, I ): f 1 ( G, I ) = f 1 ( G 0 , I 0 ) + ∂f 1 ( G 0 , I 0 ) ∂G ( G - G 0 ) + ∂f 1 ( G 0 , I 0 ) ∂I ( I - I 0 ) + h.o.t. f 2 ( G, I ) = f 2 ( G 0 , I 0 ) + ∂f 2 ( G 0 , I 0 ) ∂G ( G - G 0 ) + ∂f 2 ( G 0 , I 0 ) ∂I ( I - I 0 ) + h.o.t., where h.o.t. represents all higher order terms greater than linear Recall that f 1 ( G 0 , I 0 ) = 0 and f 2 ( G 0 , I 0 ) = 0 ( Equilibrium ). Also, g ( t ) = G ( t ) - G 0 and i ( t ) = I ( t ) - I 0 , which gives dG dt = dg dt and dI dt = di dt Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (26/68)
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Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Linearization 2 Linear Terms from Taylor’s Expansion: We carefully analyze each linear term Begin with the glucose dynamics , f 1 ( G, I ) Consider ∂f 1 ( G 0 ,I 0 ) ∂G Increases of glucose in the blood stimulates tissues to uptake glucose and liver to store glycogen Thus, this term is negative or ∂f 1 ( G 0 ,I 0 ) ∂G = - a 11 < 0 Consider ∂f 1 ( G 0 ,I 0 )
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