Clearance (pharmacology)

Diagram showing the basic physiologic mechanisms of the kidney

When referring to the function of the kidney, clearance is considered to be the amount of liquid filtered out of the blood that gets processed by the kidneys or the amount of blood cleaned per time because it has the units of a volumetric flow rate [ volume per unit time ]. However, it does not refer to a real value; "the kidney does not completely remove a substance from the total renal plasma flow."[4] From a mass transfer perspective[5] and physiologically, volumetric blood flow (to the dialysis machine and/or kidney) is only one of several factors that determine blood concentration and removal of a substance from the body. Other factors include the mass transfer coefficient, dialysate flow and dialysate recirculation flow for hemodialysis, and the glomerular filtration rate and the tubular reabsorption rate, for the kidney. A physiologic interpretation of clearance (at steady-state) is that clearance is a ratio of the mass generation and blood (or plasma) concentration.

Its definition follows from the differential equation that describes exponential decay and is used to model kidney function and hemodialysis machine function:

${\displaystyle V{\frac {dC}{dt}}=-K\cdot C+{\dot {m}}}$

(1)

Where:

• ${\displaystyle {\dot {m}}}$ is the mass generation rate of the substance - assumed to be a constant, i.e. not a function of time (equal to zero for foreign substances/drugs) [mmol/min] or [mol/s]
• t is dialysis time or time since injection of the substance/drug [min] or [s]
• V is the volume of distribution or total body water [L] or [m³]
• K is the clearance [mL/min] or [m³/s]
• C is the concentration [mmol/L] or [mol/m³] (in the United States often [mg/mL])

From the above definitions it follows that ${\displaystyle {\frac {dC}{dt}}}$ is the first derivative of concentration with respect to time, i.e. the change in concentration with time.

It is derived from a mass balance.

Clearance of a substance is sometimes expressed as the inverse of the time constant that describes its removal rate from the body divided by its volume of distribution (or total body water).

In steady-state, it is defined as the mass generation rate of a substance (which equals the mass removal rate) divided by its concentration in the blood.

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

${\displaystyle K_{el}={\dfrac {Cl_{tot}}{V_{d}}}}$

(note the usage of Cl and not Κ, not to confuse with ${\displaystyle K_{el}}$). But ${\displaystyle K_{el}}$ is also equivalent to ${\displaystyle \ln 2}$ divided by elimination rate half-life ${\displaystyle t_{1/2}}$, ${\displaystyle K_{el}={\dfrac {\ln 2}{t_{1/2}}}}$. Thus, ${\displaystyle Cl_{tot}={\dfrac {\ln 2\cdot V_{d}}{t_{1/2}}}}$ . This means, for example, that an increase in total clearance results in a decrease in elimination rate half-life, provided distribution volume is constant. Derivation of these equations can be found in e.g. Rang and Dale's Pharmacology [6]

For substances that exhibit substantial plasma protein binding, clearance is generally dependent on the total concentration (free + protein-bound) and not the free concentration.[7]

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.[7] However, the mass removal rate is the same,[7] 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.

Equation 1 is derived from a mass balance:

${\displaystyle \Delta m_{\text{body}}=(-{\dot {m}}_{\text{out}}+{\dot {m}}_{\text{in}}+{\dot {m}}_{\text{gen.}})\Delta t}$

(2)

where:

• ${\displaystyle \Delta t}$ is a period of time
• ${\displaystyle \Delta m_{\text{body}}}$ the change in mass of the toxin in the body during ${\displaystyle \Delta t}$
• ${\displaystyle {\dot {m}}_{\text{in}}}$ 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 (${\displaystyle \Delta m}$) during some time ${\displaystyle \Delta t}$ is equal to the toxin intake plus the toxin generation minus the toxin removal.

Since

${\displaystyle m_{\text{body}}=C\cdot V}$

(3)

and

${\displaystyle {\dot {m}}_{\text{out}}=K\cdot C}$

(4)

Equation A1 can be rewritten as:

${\displaystyle \Delta (C\cdot V)=\left(-K\cdot C+{\dot {m}}_{\text{in}}+{\dot {m}}_{\text{gen.}}\right)\Delta t}$

(5)

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 ${\displaystyle \Delta t}$ the result is a difference equation:

${\displaystyle {\frac {\Delta (C\cdot V)}{\Delta t}}=-K\cdot C+{\dot {m}}}$

(6)

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

${\displaystyle {\frac {d(C\cdot V)}{dt}}=-K\cdot C+{\dot {m}}}$

(7)

Using the Product Rule this can be rewritten as:

${\displaystyle C{\frac {dV}{dt}}+V{\frac {dC}{dt}}=-K\cdot C+{\dot {m}}}$

(8)

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

The general solution of the above differential equation (1) is:[8][9]

${\displaystyle C={\frac {\dot {m}}{K}}+\left(C_{o}-{\frac {\dot {m}}{K}}\right)e^{-{\frac {K\cdot t}{V}}}}$

(9)

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

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)):

${\displaystyle K={\frac {C_{U}\cdot Q}{C_{B}}}}$

(11)

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).