Two drugs, Abraxane and Taxol, are both cancer treatments, yet have differing rates at which they leave a patient’s system. Using terminology from pharmacology, Abraxane leaves the system by so-called “first-order elimination”, which means that the concentration decreases at a constant percentage rate for each unit of time that passes. Taxol leaves the system by “zero-order elimination”, which means that the concentration decreases by a constant amount for each unit of time that passes. (a) As soon as the infusion of Taxol is completed, the drug concentration in a patient’s blood is 1000 nanograms per milliliter displaystyle{left(frac{{{n}{g}}}{{{m}{l}}}right)}. 12 hours later there is displaystyle{50}frac{{{n}{g}}}{{{m}{l}}} left in the patient’s system. Use the data to construct an appr

Two drugs, Abraxane and Taxol, are both cancer treatments, yet have differing rates at which they leave a patient’s system. Using terminology from pharmacology, Abraxane leaves the system by so-called “first-order elimination”, which means that the concentration decreases at a constant percentage rate for each unit of time that passes. Taxol leaves the system by “zero-order elimination”, which means that the concentration decreases by a constant amount for each unit of time that passes. (a) As soon as the infusion of Taxol is completed, the drug concentration in a patient’s blood is 1000 nanograms per milliliter displaystyle{left(frac{{{n}{g}}}{{{m}{l}}}right)}. 12 hours later there is displaystyle{50}frac{{{n}{g}}}{{{m}{l}}} left in the patient’s system. Use the data to construct an appr

Question
Modeling data distributions
asked 2020-11-26
Two drugs, Abraxane and Taxol, are both cancer treatments, yet have differing rates at which they leave a patient’s system. Using terminology from pharmacology, Abraxane leaves the system by so-called “first-order elimination”, which means that the concentration decreases at a constant percentage rate for each unit of time that passes. Taxol leaves the system by “zero-order elimination”, which means that the concentration decreases by a constant amount for each unit of time that passes.
(a) As soon as the infusion of Taxol is completed, the drug concentration in a patient’s blood is 1000 nanograms per milliliter \(\displaystyle{\left(\frac{{{n}{g}}}{{{m}{l}}}\right)}.\) 12 hours later there is \(\displaystyle{50}\frac{{{n}{g}}}{{{m}{l}}}\) left in the patient’s system. Use the data to construct an appropriate formula modeling the blood concentration of Taxol as a function of time after the infusion is completed.
(b) As soon as the infusion of Abraxane is completed, the drug concentration in a patient’s blood is 1000 nanograms per milliliter \(\displaystyle{\left(\frac{{{n}{g}}}{{{m}{l}}}\right)}\). 24 hours later there is \(\displaystyle{50}\frac{{{n}{g}}}{{{m}{l}}}\) left in the patient’s system. Use the data to construct an appropriate formula modeling the blood concentration of Abraxane as a function of time after the infusion is completed.
(c) Find the long-term behavior of the function from part (b). Is this behavior meaningful in the context of the model?

Answers (1)

2020-11-27

(a)
Let T(t)−Amount of taxol in nanogram per mili litre present in the patient's blood at any time t in (hours) since infusion completely [Infusion completes at \(\displaystyle{t}={0}\)] As T(t) taxol follow zero order elimination.
\(\displaystyle{T}{\left({t}\right)}={m}{\left({t}\right)}+{c},\) where m is a slope
at \(\displaystyle{t}={0}\Rightarrow{T}{\left({t}\right)}={1000}\to{c}={1000}\)
and \(\displaystyle{a}{t}{t}={12}\Rightarrow{T}{\left({12}\right)}={100}\)
\(\displaystyle{50}={m}\times{12}+{1000}\)
\(\displaystyle\Rightarrow{m}=-{79.16}\)
Hence, \(\displaystyle{T}{\left({t}\right)}=-{79.16}{t}+{1000}\)
(b)
\(A(t)\)to Amount of Abraxane in ng/ml in patient's bloodat any time t in (hours) since infusion. As Abrasane follows Istorder elimination.
\(\displaystyle{A}{\left({t}\right)}={P}{\left(\theta\right)}^{t}\)
at \(\displaystyle{t}={0},{A}{\left({t}\right)}={1000}\Rightarrow{1000}={P}{\left(\theta\right)}^{0}\to{P}={1000}\)
Now,
\(\displaystyle{A}{\left({t}\right)}={1000}{\left(\theta\right)}^{t}At t = 24, A(t) = 50\)
\(\displaystyle{50}={1000}{\left(\theta\right)}^{24}\)
\(\displaystyle\theta={\left(\frac{1}{{20}}\right)}^{{\frac{1}{{24}}}}\Rightarrow\theta={0.88265}\)
Hence, \(\displaystyle{A}{\left({t}\right)}={1000}{\left({0.8826}\right)}^{t}\)
(c)
From the part (b): \(\displaystyle{A}{\left({t}\right)}={1000}{\left({0.8826}\right)}^{t}\ldots\ldots{\left({1}\right)}\)
where,
\(\displaystyle{A}{\left({t}\right)}=\) Amount of Abraxane in the patient's blood.
The equation (1) shows that the amount of Abraxane inpatient'sblood is exponentially decreasing in the patient'sblood.

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