Question

We want to calculate the average concentration of a relatively short acting drug in a patient. A bolus of the drug is given quickly via intravenous (IV) and the initial concentration of the drug in the patient is 20 mg/dL. The drug has a half-life of 2 hours. What is the average concentration of the drug in this patient's body over 5 half-lives? Assume first-order kinetics.

Answer 1

Ans. First order kinetics-

ln ([A]_t / [A]₀) = -kt                                -equation 1

where, [A]_t = Final concentration

            [A]₀ = Initial concentration

            k = rate constant

            t = time of reaction

Also, Half-life of first order reaction, t_½ = 0.693 / k

            Or, k = 0.693 / t_½                                        - equation 2

Combining equation 1 and 2 –

            ln ([A]_t / [A]₀) = -(0.693 / t_½) t               

            or, ln [A]_t – ln [A]₀ = -(0.693 / t_½) t                    - equation 3

Given-

            Initial [drug], [A]₀ = 20 mg/ dL

            Time, t = 5 x (half-life) = 5 x 2 hrs = 10 hrs

Now,

Putting the values in equation 3-

            ln [A]_t – ln (20) = -(0.693 / 2 hr ) x 10 hr

            Or, ln [A]_t – 2.9957 = - 3.465

            Or, 2.303 log [A]_t = - 3.465 + 2.9957 = -0.4693

            Or, log [A]_t = -0.4693 / 2.303 = -0.203778

            Or, [A]_t = antilog (- 0.203778) = 0.625

Therefore, final [Drug] in patient’s body, [A]_t = 0.625 mg/ dL