Medicine core problem solutions

Solution to Problem 1 – Drug resonance half-life#

$$ t_{1/2} = \frac{\ln 2}{ΛΘ}. $$

If $$Θ$$ decreases by 10%:

$$ Θ' = 0.9Θ, \quad t_{1/2}' = \frac{\ln 2}{Λ(0.9Θ)} = \frac{t_{1/2}}{0.9}. $$

Answer: Half-life increases by 11.1%.

Solution to Problem 2 – Heart rhythm resonance#

$$ R = T_f D_3. $$

If $$T_f$$ increases by 8%:

$$ T_f' = 1.08 T_f, \quad R' = 1.08 R. $$

Answer: Heart rhythm increases by 8%.

Solution to Problem 3 – Dose-response resonance#

$$ S = \frac{X}{1 + e^{-τ_r}}. $$

As $$τ_r$$ increases:

• $$e^{-τ_r}$$ decreases,
• denominator decreases,
• $$S$$ increases.

Answer: Response increases monotonically with $$τ_r$$.