# Two drugs, Abraxane and Taxol, are both cancer treatments, yet have differing rates at which they leave a patient’s system. Using terminology from pha

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 $\left(\frac{ng}{ml}\right).$ 12 hours later there is $50\frac{ng}{ml}$ 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 $\left(\frac{ng}{ml}\right)$. 24 hours later there is $50\frac{ng}{ml}$ 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?
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(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 $t=0$] As T(t) taxol follow zero order elimination.
$T\left(t\right)=m\left(t\right)+c,$ where m is a slope
at $t=0⇒T\left(t\right)=1000\to c=1000$
and $att=12⇒T\left(12\right)=100$
$50=m×12+1000$
$⇒m=-79.16$
Hence, $T\left(t\right)=-79.16t+1000$
(b)
$A\left(t\right)$to Amount of Abraxane in ng/ml in patient's bloodat any time t in (hours) since infusion. As Abrasane follows Istorder elimination.
$A\left(t\right)=P{\left(\theta \right)}^{t}$
at $t=0,A\left(t\right)=1000⇒1000=P{\left(\theta \right)}^{0}\to P=1000$
Now,
$A\left(t\right)=1000{\left(\theta \right)}^{t}Att=24,A\left(t\right)=50$
$50=1000{\left(\theta \right)}^{24}$
$\theta ={\left(\frac{1}{20}\right)}^{\frac{1}{24}}⇒\theta =0.88265$
Hence, $A\left(t\right)=1000{\left(0.8826\right)}^{t}$
(c)
From the part (b): $A\left(t\right)=1000{\left(0.8826\right)}^{t}\dots \dots \left(1\right)$
where,
$A\left(t\right)=$ 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.