Applications of the Logistic Curve
The logistic curve has applications in many fields. In medicine, for example, it can represent the amount of a medication that is in the bloodstream for a short period of time just after ingestion while the drug is filtered into the bloodstream. A person ingests 200 milligrams (mg) of a medication. After 30 minutes, there are 100 mg of the medication in the bloodstream. After 60 minutes, there are 180 mg of medication in the bloodstream. Using the spreadsheet, assign appropriate values to L and C and then adjust B until you are able to match the given infomation.
Part A: L =
Part B: C=
Part C: B=
Part D: The amount of medication in the bloodstream after 20 minutes is approximately how many mg?
Part A: L represents the maximum capacity or limit of the logistic curve. Since we don’t have any information about the maximum amount of medication that can be in the bloodstream, we cannot assign a value to L.
Part B: C represents the midpoint or the time at which the curve reaches half of its maximum value. We can calculate C by finding the time it takes for the medication to reach half of its initial value:
100 = L / (1 + e^(-B*(30-C))) 180 = L / (1 + e^(-B*(60-C)))
Dividing the second equation by the first one, we get:
1.8 = e^(30B)
Taking the natural logarithm of both sides, we get:
ln(1.8) = 30B
B = ln(1.8)/30 ≈ 0.0231
Now we can use one of the two equations above to solve for C:
100 = L / (1 + e^(-0.0231*(30-C))) C ≈ 29.05
Therefore, we can assign the value of 29.05 to C.
Part C: B represents the growth rate of the curve. We have already calculated B in Part B, which is approximately 0.0231.
Part D: To estimate the amount of medication in the bloodstream after 20 minutes, we can use the logistic curve equation:
M(t) = L / (1 + e^(-B*(t-C)))
where t is the time in minutes and M(t) is the amount of medication in milligrams.
Using the values of B and C calculated above, we get:
M(20) = L / (1 + e^(-0.0231*(20-29.05)))
However, we still don’t know the value of L. Since we don’t have any information about the maximum amount of medication that can be in the bloodstream, we cannot estimate M(20).
