Biopharmaceutical Equations

1. Henderson-Hasselbalch Equation:

   Calculates the pH of a buffer solution and predicts how pH changes affect drug solubility and ionization.

   $$ \text{pH} = \text{p}K_a + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) $$

2. Michaelis-Menten Equation:

   Describes enzyme kinetics, particularly enzyme-substrate binding and the rate of product formation.

   $$ V_0 = \frac{V_{\max} [S]}{K_m + [S]} $$

3. First-Order Kinetics Equation:

   Describes the rate of drug elimination from the body when the elimination rate is proportional to the drug concentration.

   $$ \frac{d[A]}{dt} = -k[A] $$

4. Zero-Order Kinetics Equation:

   Describes the rate of drug elimination from the body when the elimination rate is constant.

   $$ \frac{d[A]}{dt} = -k $$

5. Volume of Distribution (Vd) Equation:

   Describes the apparent volume in which a drug is distributed in the body relative to its plasma concentration.

   $$ V_d = \frac{\text{Amount of drug in body}}{\text{Concentration of drug in plasma}} $$

6. Bioavailability (F) Equation:

   Measures the fraction of the administered drug that reaches systemic circulation unchanged.

   $$ F = \left( \frac{\text{AUC}_{\text{oral}}}{\text{Dose}_{\text{oral}}} \right) \times \left( \frac{\text{Dose}_{\text{IV}}}{\text{AUC}_{\text{IV}}} \right) $$

7. Clearance (CL) Equation:

   Describes the rate at which a drug is removed from the body.

   $$ CL = \frac{\text{Rate of elimination}}{\text{Concentration of drug in plasma}} $$

8. Half-Life (t½) Equation:

   Describes the time it takes for the drug concentration in the plasma or the amount of drug in the body to decrease by half.

   $$ t_{1/2} = \frac{0.693 \cdot V_d}{CL} $$

9. Steady-State Concentration (Css) Equation:

   Describes the concentration of drug achieved when the rate of drug input equals the rate of drug elimination.

   $$ C_{ss} = \frac{\text{Rate of drug input}}{\text{Rate of drug elimination}} $$

10. AUC (Area Under the Curve) Equation:

    Measures the total exposure to a drug over time, typically used to assess drug bioavailability.

    $$ \text{AUC} = \int C(t) \, dt $$

11. Exponential Decay Equation:

    Describes drug concentration over time, typically used in pharmacokinetics to model drug elimination.

    $$ cp = cp_0 \cdot e^{-kt} $$

    • \(cp\): Drug concentration at time \(t\).
    • \(cp_0\): Initial drug concentration.
    • \(k\): Elimination rate constant.
    • \(t\): Time.
    • \(e\): Base of the natural logarithm (approximately 2.71828).
12. Loading Dose Equation:

    Calculates the initial dose of a drug required to rapidly achieve a desired drug concentration in the body.

    $$ D_{\text{load}} = C_{\text{target}} \times V_d $$

13. Maintenance Dose Equation:

    Calculates the dose of a drug required to maintain a desired drug concentration in the body over time.

    $$ D_{\text{maintain}} = \frac{C_{\text{target}} \times CL \times \tau}{F} $$

14. Accumulation Equation:

    Describes the accumulation of a drug in the body over multiple dosing intervals.

    $$ R = \frac{1}{1 - e^{-k \tau}} $$

    • \(R\): Accumulation factor.
    • \(k\): Elimination rate constant.
    • \(\tau\): Dosing interval.
15. Maximum Concentration (C_max) Equation:

    Describes the peak concentration that a drug achieves in the bloodstream after administration.

    $$ C_{\text{max}} = \frac{F \cdot D}{V_d \cdot k} \left( 1 - e^{-k \tau} \right) $$

16. Minimum Concentration (C_min) Equation:

    Describes the trough concentration that a drug achieves in the bloodstream just before the next dose.

    $$ C_{\text{min}} = C_{\text{max}} \cdot e^{-k \tau} $$

17. IV Bolus Dose Equation:

    Describes the concentration of a drug in the plasma after an intravenous bolus dose.

    $$ C(t) = \frac{D_{\text{IV}}}{V_d} e^{-kt} $$

18. Oral Tablet Dose Equation:

    Describes the concentration of a drug in the plasma after an oral tablet dose.

    $$ C(t) = \frac{F D_{\text{oral}}}{V_d} \frac{k_a}{k_a - k} \left( e^{-kt} - e^{-k_a t} \right) $$

19. Plasma Volume (Vp) Equation:

    Describes the volume of the plasma compartment.

    $$ V_p = \frac{D}{C_p} $$

20. Tissue Volume (Vt) Equation:

    Describes the volume of the tissue compartment.

    $$ V_t = \frac{D \cdot f_t}{C_t} $$

21. Binding Volume of Distribution (Vd_b) Equation:

    Describes the volume of distribution considering drug binding.

    $$ (V_d)_b = \frac{V_p + V_t \cdot \frac{f_t}{f_p}}{1 + \frac{f_t}{f_p}} $$

22. Cockcroft-Gault Method Equation:

    Estimates creatinine clearance for drug dosing adjustments in renal impairment.

    $$ \text{CrCl} = \frac{(140 - \text{age}) \times \text{weight} \times (0.85 \text{ if female})}{72 \times \text{serum creatinine}} $$