3. [Total: 19 pts] Blood glucose and insulin regulation The equations that characterize the steady state regulation of glucose and insulin can be derived by taking into account the main factors

that affect the appearance and disappearance of each of these substances in the body. The mass balance for blood glucose (x) is given by: Q₁ = λx +vxy (Eqn. 1) where Q₁ represents the flow-rate (assume to be constant) at which glucose enters the blood through absorption from the gastrointestinal tract or through production from the liver, xx represents the rate at which the body tissues utilize glucose without help from insulin, and vxy is the rate at which glucose is metabolized with facilitation from insulin. The blood concentration of insulin is represented by y, and λ and v are constant parameters. Thus, Eqn. 1 represents the "glucose response to insulin". By applying similar mass-balance considerations to the production of insulin from the pancreas and the subsequent destruction of insulin, we can derive the following expression for the "insulin response to glucose": (Egn. 2a) (Eqn. 2b) where and are constant parameters. Insulin is not produced when glucose is lower than the threshold parameter (Eqn.2a). Note that the insulin production and destruction rates are combined into one parameter, (zeta). By using empirically derived values for the model parameters (Q₁, A, v, 3, 4) in Egns. 1 and 2a/b, these equations allow us to predict the blood concentrations of glucose (x) and insulin (y) under various conditions. The figure below displays the plots corresponding to these responses in normal and diabetic subjects. Point N is the operating point of a normal subject. B с D Insulin concentration (mu.mL-¹) 0.2 0.15 y = 0, x≤0 y = {(x-4), x> 0.1 0.05 0 0.2 0.4 0.6 0.8 N 1 1.2 1.4 1.6 1.8 2/na) [2 pts] Estimate from the figure the operating values of glucose and insulin in the blood of a normal subject. b) [3 pts] Suppose this person eats junk food everyday and does not exercise, and after many years, develops "insulin resistance" (i.e., the body requires more and more insulin to metabolize the same amount of glucose). Assuming his pancreas still functions normally. Which parameter(s) need to change to simulate this condition? Determine his new operating values of glucose and insulin. Label the new operating point "b". Briefly explain how you arrived at your answers. c) [3 pts] After many years of untreated insulin resistance, this subject's pancreas can now operate at half of its full capacity. Which parameter(s) need to change to simulate this condition? Determine his new operating values of glucose and insulin. Label the new operating point "c". Briefly explain how you arrived at your answers./nd) [5 pts] The subject does nothing to change his lifestyle and the pancreas eventually fails. Suppose that his pancreas is removed and replaced by a machine that continuously infuses insulin into his body independent of the glucose level. The machine is adjusted such that it infuses the appropriate amount of insulin, restoring the plasma glucose concentration to the normal level. Sketch as accurately as possible in the provided figure to reflect the described condition. Label the curve(s) "d". Briefly explain how you arrived at your answer. e) [3 pts] From the provided figure, is it possible for you to estimate the parameter of a normal healthy person? If so, provide your best estimate of pp. Briefly explain how you arrived at your answer. f) [3 pts] From the provided figure, is it possible for you to estimate the parameter < (Eqn. 2b) of a normal healthy person? If so, provide your best estimate of and from which graph. Briefly explain how you arrived at your answer.