Mathematical equations for sigma-minus and rate excretion method to determine various PK parameters

The Sigma-minus method and the Rate excretion method are classical pharmacokinetic (PK) approaches used to analyze drug excretion and elimination. These methods help determine parameters like elimination rate constant ($k_e$

   

half-life, clearance, and others. Below are the key equations for both methods. 

Sigma-minus Method

The sigma-minus method is used to calculate PK parameters by plotting the cumulative amount of drug excreted.

Key Equations:

1. Amount of Drug Excreted in Urine:

   $$A_e=D\times F\times (1-e^{-k_e\cdot t})$$

   Where:

   • $A_e$: Amount of drug excreted in urine.
   • $D$: Administered dose.
   • $F$: Bioavailability.
   • $k_e$: Elimination rate constant.
   • $t$: Time after drug administration.

2. Sigma-minus Plot:

   $$(A_e{)}_{\mathrm{\infty }}-A_e(t)=A_e(t)\cdot e^{-k_e\cdot t}$$
   • Plot $(A_e{)}_{\mathrm{\infty }}-A_e(t)$ (cumulative amount excreted vs. time).
   • The slope of the line is proportional to the elimination rate constant $k_e$.

Rate Excretion Method

This method involves analyzing the rate at which a drug is excreted in the urine over time. It is based on the first-order elimination kinetics model.

Key Equations:

1. Rate of Excretion:

   $$\frac{d{A}_e}{dt}=k_e\cdot A_t$$

   Where:

   • $\frac{d{A}_e}{dt}$: Rate of drug excretion at time $t$.
   • $A_t$: Amount of drug in the body at time $t$.
   • $k_e$: Elimination rate constant.

2. Logarithmic Transformation:

   $$\log \left(\frac{d{A}_e}{dt}\right)=\log (k_e\cdot D\cdot F)-\frac{k_e\cdot t}{2.303}$$

   This equation is often used to generate a linear plot, from which $k_e$ can be derived from the slope.

Half-life (t₁/₂):

$$t_{1\mathrm{/}2}=\frac{0.693}{k_e}$$

Total Body Clearance (Cl):

$$Cl=\frac{D\cdot F}{AU{C}_{\mathrm{\infty }}}$$

Where $AU{C}_{\mathrm{\infty }}$ is the area under the plasma concentration-time curve from 0 to infinity.

PK Parameters Determined:

1. Elimination rate constant $(k_e)$
2. Half-life $(t_{1\mathrm{/}2})$
3. Total clearance $(Cl)$
4. Volume of distribution $(V_d)$
5. Bioavailability $(F)$

These methods are used in non-compartmental analysis for the interpretation of urine excretion data and are particularly useful when intravenous data is unavailable.