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Problem: Determine the insulin flow rates in the human body

A significant amount of diabetic patients use daily insulin injections to keep their blood sugar stable. Increasing the amount of insulin in the blood decreases the amount of sugar in the blood. Several methods exist for monitoring blood sugar, however, the inability to achieve a predictable blood sugar over time is a frustrating problem that constantly plagues diabetics.

The model shown in Figure 1 represents the flow of insulin through specific parts of the body. For analytical purposes, this model does not take into account other factors, such as glucose, that may normally affect the concentration of insulin throughout the body. Using this model, it is your goal to predict the level of insulin in the blood over time after an insulin injection.

Figure 1 shows the flow of insulin in the body. A patient is first given an insulin injection, usually in the abdomen or upper arm. The insulin then dissolves into the bloodstream. Once in the blood, insulin flows to and from the kidneys, as well as to and from the pancreas.

Let x₁(t) be the concentration of insulin in the blood.
Let x₂(t) be the concentration of insulin in the kidneys.
Let x₃(t) be the concentration of insulin in the pancreas.
Let x₄(t) be the concentration of insulin in the abdomen.
Initialize x₄(0) = 25 Units (U) to be the amount of insulin initially present in the abdomen due to the injection.

Let k_in = 2 U/hr be the flow rate of insulin from the abdomen into the blood.
Let k₁₂ = 1 U/hr be the flow rate of insulin from the blood into the kidneys.
Let k₂₁ = 1.5 U/hr be the flow rate of insulin from the kidneys into the blood.
Let k₁₃ = 2 U/hr be the flow rate of insulin from the blood into the pancreas.
Let k₃₁ = 1.75 U/hr be the flow rate of insulin from the pancreas into the blood.

Figure 1

Objective:

1. Using the model, determine the flow rate equations for this system.
2. Solve these equations for the concentration of insulin in each compartment.
3. Plot the concentrations of insulin in each compartment for up to 10 hours.
4. Given these results, estimate the steady-state level of insulin in the blood and the time at which this level is reached.

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