Question

Suppose a patient is given a continuous intravenous infusion of glucose at a constant rate of r mg/mm. Then, the rate at which the amount of glucose in the bloodstream is changing at time t (in minutes) because of this infusion is given by

A'(t) = re⁻ᵃᵗ
mg/min, where a is a positive constant associated with the rate at which excess glucose is eliminated from the bloodstream and is dependent on the patient's metabolism rate.
Derive an expression for the amount of glucose in the bloodstream at time t if A(0) = 0

Answer 1

The amount of glucose in the bloodstream at time t is given by A(t) = (r/a)(1 - e⁻⁰ᵗ), by integrating the rate of the glucose change in the bloodstream and using the initial condition A(0) = 0.

Step-by-step explanation:

The student is asking for an expression for the amount of glucose in the bloodstream at time t, given a continuous intravenous infusion of glucose.

To find the amount of glucose (A(t)), we must integrate the rate of change of the amount of glucose (A'(t)) with respect to time.

Given that A'(t) = re⁻⁰ᵗ mg/min, the integral of this with respect to t from 0 to t provides us the total glucose amount at time t, assuming that A(0) = 0.

Therefore, the integral of A'(t) with respect to time is:

• ∫ A'(t) dt = ∫ re⁻⁰ᵗ dt
• A(t) = -öᵗr/a + C
• Since A(0) = 0, we find C by plugging in the initial condition, resulting in C = r/a
• Thus, A(t) = (r/a)(1 - e⁻⁰ᵗ)

This formula represents the amount of glucose in the bloodstream at time t as a function of the infusion rate r and the elimination rate constant a.