# Clearance (medicine)

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In medicine, the clearance, also renal clearance or renal plasma clearance (when referring to the function of the kidney), of a substance is 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.

It 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 / time ]. However, it does not refer to a real value; "[t]he kidney does not completely remove a substance from the total renal plasma flow."[1] From a mass transfer perspective[2] 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:

${\displaystyle V{\frac {dC}{dt}}=-K\cdot C+{\dot {m}}\qquad (1)}$

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 USA 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.

## Derivation of equation

Equation 1 is derived from a mass balance:

${\displaystyle \Delta m_{body}=(-{\dot {m}}_{out}+{\dot {m}}_{in}+{\dot {m}}_{gen.})\Delta t\qquad (2)}$

where:

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

${\displaystyle m_{body}=C\cdot V\qquad (3)}$

and

${\displaystyle {\dot {m}}_{out}=K\cdot C\qquad (4)}$

Equation A1 can be re-written as:

${\displaystyle \Delta (C\cdot V)=(-K\cdot C+{\dot {m}}_{in}+{\dot {m}}_{gen.})\Delta t\qquad (5)}$

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

${\displaystyle {\frac {\Delta (C\cdot V)}{\Delta t}}=-K\cdot C+{\dot {m}}\qquad (6)}$

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

${\displaystyle {\frac {d(C\cdot V)}{dt}}=-K\cdot C+{\dot {m}}\qquad (7)}$

Using the chain rule this can be re-written as:

${\displaystyle C{\frac {dV}{dt}}+V{\frac {dC}{dt}}=-K\cdot C+{\dot {m}}\qquad (8)}$

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

${\displaystyle V{\frac {dC}{dt}}=-K\cdot C+{\dot {m}}\qquad (1)}$

## Solution to the differential equation

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

${\displaystyle C={\frac {\dot {m}}{K}}+(C_{o}-{\frac {\dot {m}}{K}})e^{-{\frac {K\cdot t}{V}}}\qquad (9)}$[3][4]

Where:

• Co 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

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

${\displaystyle C_{\infty }={\frac {\dot {m}}{K}}\qquad (10a)}$

The above equation (10a) can be re-written as:

${\displaystyle K={\frac {\dot {m}}{C_{\infty }}}\qquad (10b)}$

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)):

${\displaystyle K={\frac {C_{U}\cdot Q}{C_{B}}}\qquad (11)}$

Where:

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

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).