Clearance (pharmacology)

Clearance, half-life and volume of distribution

There is an important relationship between clearance, elimination half-life and distribution volume. The elimination rate constant of a drug ${\displaystyle K_{\text{el}}}$ is equivalent to total clearance divided by the distribution volume

${\displaystyle K_{\text{el}}={\displaystyle \frac{C{l}_{\text{tot}}}{V_{\text{d}}}}}$

(note the usage of Cl and not Κ, not to confuse with ${\displaystyle K_{\text{el}}}$). But ${\displaystyle K_{\text{el}}}$ is also equivalent to ${\displaystyle \ln 2}$ divided by elimination rate half-life ${\displaystyle t_{1/2}}$, ${\displaystyle K_{\text{el}}={\displaystyle \frac{\ln 2}{t_{1/2}}}}$. Thus, ${\displaystyle C{l}_{\text{tot}}={\displaystyle \frac{\ln 2\cdot V_{\text{d}}}{t_{1/2}}}}$ . This means, for example, that an increase in total clearance results in a decrease in elimination rate half-life, provided distribution volume is constant.

Effect of plasma protein binding

For substances that exhibit substantial plasma protein binding, clearance is generally dependent on the total concentration (free + protein-bound) and not the free concentration.

Most plasma substances have primarily their free concentrations regulated, which thus remains the same, so extensive protein binding increases total plasma concentration (free + protein-bound). This decreases clearance compared to what would have been the case if the substance did not bind to protein. However, the mass removal rate is the same, because it depends only on concentration of free substance, and is independent on plasma protein binding, even with the fact that plasma proteins increase in concentration in the distal renal glomerulus as plasma is filtered into Bowman's capsule, because the relative increases in concentrations of substance-protein and non-occupied protein are equal and therefore give no net binding or dissociation of substances from plasma proteins, thus giving a constant plasma concentration of free substance throughout the glomerulus, which also would have been the case without any plasma protein binding.

In other sites than the kidneys, however, where clearance is made by membrane transport proteins rather than filtration, extensive plasma protein binding may increase clearance by keeping concentration of free substance fairly constant throughout the capillary bed, inhibiting a decrease in clearance caused by decreased concentration of free substance through the capillary.

Derivation of equation

Equation 1 is derived from a mass balance:

where:

• ${\displaystyle \mathrm{\Delta }t}$ is a period of time
• ${\displaystyle \mathrm{\Delta }{m}_{\text{body}}}$ the change in mass of the toxin in the body during ${\displaystyle \mathrm{\Delta }t}$
• ${\displaystyle {\dot{m}}_{\text{in}}}$ is the toxin intake rate
• ${\displaystyle {\dot{m}}_{\text{out}}}$ is the toxin removal rate
• ${\displaystyle {\dot{m}}_{\text{gen.}}}$ is the toxin generation rate

In words, the above equation states: "The change in the mass of a toxin within the body (${\displaystyle \mathrm{\Delta }m}$) during some time ${\displaystyle \mathrm{\Delta }t}$ is equal to the toxin intake plus the toxin generation minus the toxin removal."

Since

and

Equation A1 can be rewritten as:

If one lumps the in and gen. terms together, i.e. ${\displaystyle \dot{m}={\dot{m}}_{\text{in}}+{\dot{m}}_{\text{gen.}}}$ and divides by ${\displaystyle \mathrm{\Delta }t}$ the result is a difference equation:

If one applies the limit ${\displaystyle \mathrm{\Delta }t\to 0}$ one obtains a differential equation:

Using the product rule this can be rewritten as:

If one assumes that the volume change is not significant, i.e. ${\displaystyle C\frac{dV}{dt}=0}$, the result is Equation 1:

Solution to the differential equation

The general solution of the above differential equation (1) is:

Where:

• C_o is the concentration at the beginning of dialysis or the initial concentration of the substance/drug (after it has distributed) [mmol/L] or [mol/m³]
• e is the base of the natural logarithm

Steady-state solution

The solution to the above differential equation (9) at time infinity (steady state) is:

The above equation (10a) can be rewritten as:

The above equation (10b) makes clear the relationship between mass removal and clearance. It states that (with a constant mass generation) the concentration and clearance vary inversely with one another. If applied to creatinine (i.e. creatinine clearance), it follows from the equation that if the serum creatinine doubles the clearance halves and that if the serum creatinine quadruples the clearance is quartered.

Measurement of renal clearance

Renal clearance can be measured with a timed collection of urine and an analysis of its composition with the aid of the following equation (which follows directly from the derivation of (10b)):

Where:

• K is the clearance [mL/min]
• C_U is the urine concentration [mmol/L] (in the USA often [mg/mL])
• Q is the urine flow (volume/time) [mL/min] (often [mL/24 h])
• C_B is the plasma concentration [mmol/L] (in the USA often [mg/mL])

When the substance "C" is creatinine, an endogenous chemical that is excreted only by filtration, the clearance is an approximation of the glomerular filtration rate. Inulin clearance is less commonly used to precisely determine glomerular filtration rate.

Note - the above equation (11) is valid only for the steady-state condition. If the substance being cleared is not at a constant plasma concentration (i.e. not at steady-state) K must be obtained from the (full) solution of the differential equation (9).