A first-order process is a process that has a rate proportional to the amount of the reactant involved in this process.

For example, if the rate of the process is proportional to the amount of the drug available for the process, this means that this process follows first-order process. The proportionality constant is the first-order rate constant for the process. e.g.

Elimination rate = Elimination rate constant x drug amount

Transfer rate = Transfer rate constant x drug amount

Distribution rate = Distribution rate constant x drug amount

X and Y : The amount of the drug in the central and peripheral compartments, respectively (amount).

k₁₂ and k₂₁ : The first-order transfer rate constant between the central and the peripheral compartments (time^-1).

k₁₀ : The first-order elimination rate constant (time^-1).

α : The hybrid first-order rate constant for the distribution process (time^-1)

β : The hybrid first-order rate constant for the elimination process (time^-1)

α_{t 1/2} and β_{t 1/2} : half lives for the distribution process and the elimination process, respectively (time)

V_c: The volume of distribution of the central compartment (volume).

Vd_ss : The volume of distribution at steady state (volume).

V_β or Vd_area : The volume of distribution during the elimination phase (volume).

dx/dt is the rate of change of the amount of the drug in the central compartment (amount/time).

X is the amount of the drug in the central compartment (amount).

Y is the amount of the drug in the peripheral compartment (amount).

k₁₂ is the first-order transfer rate constant from the central to the peripheral compartment (time^-1).

k₂₁ is the first-order transfer rate constant from the peripheral to the central compartment (time^-1).

k₁₀ is the first-order elimination rate constant (time^-1).

dY/dt is the rate of change of the amount of the drug in the peripheral compartment (amount/time).

X is the amount of the drug in the central compartment (amount).

Y is the amount of the drug in the peripheral compartment (amount).

k₁₂ is the first-order transfer rate constant from the central to the peripheral compartment (time^-1).

k₂₁ is the first-order transfer rate constant from the peripheral to the central compartment (time^-1).

Derivation of the integrated equation that describes the plasma conc-time profile

The differential equation that describes the rate of change of the amount of the drug in the central compartment is:

Integration of this differential equation requires the use of laplace transform.

Integration yields:

The plasma conc-time profile in the central compartment can be described by:

As a result of the integration the following two relationship are obtained:

where

and

Cp is the plasma concentration (amount/volume).

D is the dose of the drug (amount).

V_c is the volume of the central compartment (volume).

k₂₁ is the first-order transfer rate constant from the peripheral to the central compartment (time^-1).

α is the hybrid first-order rate constant for the distribution process (time)

β is the hybrid first-order rate constant for the elimination process (time^-1)

t is the time (time).

Cp is the plasma concentration (amount/volume).

α is the hybrid first-order rate constant for the distribution process (time^-1).

β is the hybrid first-order rate constant for the elimination process (time^-1).

A and B are the hybrid coefficients (amount/volume)

A

is equal to    

D(α-k₂₁)

V_c(α-β)

B

is equal to    

D(k₂₁-β)

V_c(α-β)

t is the time (time).