Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of

Linear applications of systems of 1 st order des

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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 Tolerance Test 1 Glucose Tolerance Test (GTT) and Ackerman Model GTT Patient fasts for 12 hours Patient drinks 1.75 mg of glucose/kg of body weight Glucose levels in blood is monitored for 4-6 hours Ackerman Model Compartmental model for glucose and insulin in the body Model tracks glucose in the blood Model given by equation G ( t ) = G 0 + Ae - αt cos( ω ( t - δ )) 5 parameters fit to GTT blood data Use parameters α and ω to detect diabetes Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (32/68)
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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 Tolerance Test 2 Data for a Normal Subject A and Diabetic Subject B t (hr) A B t (hr) A B 0 70 100 2 75 175 0.5 150 185 2.5 65 105 0.75 165 210 3 75 100 1 145 220 4 80 85 1.5 90 195 6 75 90 Model for Normal Patient with best parameters G 1 ( t ) = 79 . 2 + 171 . 5 e - 0 . 99 t cos(1 . 81( t - 0 . 901)) Model for Diabetic Patient with best parameters G 2 ( t ) = 95 . 2 + 263 . 2 e - 0 . 63 t cos(1 . 03( t - 1 . 52)) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (33/68)
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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 Tolerance Test 3 Graph of data and models 0 1 2 3 4 5 6 0 50 100 150 200 250 t (hr) Glucose (mg/dl) GTT Model Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (34/68)
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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 Tolerance Test 4 Model for Normal Patient with best parameters is G 1 ( t ) = 79 . 2 + 171 . 5 e - 0 . 99 t cos(1 . 81( t - 0 . 901)) Calculus techniques show a maximum at t max = 0 . 624 hr with G 1 ( t max ) = 160 . 3 ng/dl and a minimum at t min = 2 . 360 hr with G 1 ( t min ) = 64 . 7 ng/dl Model for Diabetic Patient with best parameters is G 2 ( t ) = 95 . 2 + 263 . 2 e - 0 . 63 t cos(1 . 03( t - 1 . 52)) , Similar calculations give the maximum at t max = 0 . 987 hr with G 2 ( t max ) = 215 . 8 ng/dl and a minimum at t min = 4 . 037 hr with G 2 ( t min ) = 77 . 6 ng/dl Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (35/68)
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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 Tolerance Test 5 The Ackerman Test examines the natural frequency , ω 0 , (study in next chapter) and period, T 0 , of the models, where ω 2 0 = α 2 + ω 2 and T 0 = 2 π ω 0 Our models give the normal subject ω 0 = 2 . 067 and T 0 = 3 . 04 hr and the diabetic subject ω 0 = 1 . 210 and T 0 = 5 . 19 hr Note: T 0 > 4 suggests diabetes Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (36/68)
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Introduction Linear Applications of Systems of 1 st Order DEs
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