(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.