One Compartment Model

Adapted from Physics, Pharmacology and Physiology for Anaesthetists

• Where:
• C₀ = the 'outside compartment'
• C₁ = central compartment
• K₀₁ = rate constant for a drug moving from C₀ to C₁ i.e. rate of administration of the drug
• K₁₀ = rate constant of elimination for a drug moving from C₁ to C₀
• V_D = volume of distribution

• The model is described by a negative exponential curve of plasma concentration vs. time
• The equation for this in a one compartment model is:

C = C₀.e^-kt

• Where:
• C = drug concentration and is the dependent variable
• C₀ = the initial drug concentration (concentration at t = 0)
• t = time and is the independent variable
• k = the rate constant for elimination


• The rate constant 'k' (units: 1/min) is the fraction of the volume of distribution from which drug is removed per unit time
• The actual volume of from which drug is removed per unit time is clearance (ml/min)
• As such, clearance is a product of the rate constant 'k' and the volume of distribution: Cl = k.V_D
• We know that the time constant (τ) is 1/k, which can be substituted into the equation making: Cl = V_D/τ


• For any given one compartment model, the volume of distribution and time constant are both constant
• As clearance is the ratio of these two constants, clearance in a one compartment model must also be constant

• Taking the equation for the one compartment model:

C = C₀.e^-kt

• One can then take natural logarithms (Ln) of both sides, and simplify the equation:

C = C₀.e^-kt

Ln(C) = Ln(C₀.e^-kt)

Ln(C) = Ln(C₀) + Ln(e^-kt)

Ln(C) = Ln(C₀) + (-kt x Ln(e))

Ln(C) = Ln(C₀) -kt

• This equation describes a linear graph of the form y = mx + c
• The gradient of the line (m) is represented by '-k'
• The y-intercept of the line (c) is represented by 'Ln(C₀)'
• As such, a semi-logarithmic plot of ln(concentration) vs. time is linear:

• If we know the dose of a drug given and its concentration at time zero (C₀), we can work out its volume of distribution: V_D = dose / C₀
• If we know the volume of distribution, following a single dose of a drug its concentration at time zero is therefore: C₀ = dose/V_D


• The amount of drug left in the body declines as a negative exponent in the same way that concentration does: X_t = X.e^-kt
• Therefore the rate of elimination of the drug (in mg/min) is the tangent to this curve, which is found by the differential of X_t and t i.e.

dX_t/dt = -kX_t



• We know from above that clearance is a product of the rate constant for elimination and volume of distribution (Cl = k.V_D)
• This can be rearranged to show k = Cl/V_D, which we can substitue into the above equation to give:

dX_t/dt = -(Cl/V_D).X_t

• This can be re-arranged to give:

dX_t/dt = -Cl(X_t/V_D)



• As X_t represents an amount of drug at a given time (mg) and V_D is a volume (ml), this gives the drug concentration (mg/ml)
• The equation thus becomes:

dX_t/dt = -Cl.C

• I.e. the rate of elimination of the drug is equal to the clearance of the drug multiplied by its concentration

• In a one-compartment model, it is possible to calculate the rate of infusion required in order to maintain a given plasma concentration
• In order to maintain a steady plasma concentration, the rate of infusion must equal the rate of elimination
• As above, the rate of elimination is a product of the concentration of a drug and its clearance (C.Cl)
• Therefore, rate of steady state infusion = desired concentration (mg/ml) x clearance (ml.min^-1)