Clearance (medicine)
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219884087
2008-06-17T08:41:18Z
Chaldor
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more correctly defined all methods of substance secretion/reabsorption in introduction
In [[medicine]], the '''clearance''' is a measurement of the renal [[excretion]] ability. Although clearance may also involve other organs than the kidney, it is almost synonymous with '''renal clearance''' or '''renal plasma clearance'''. Each substance has a specific clearance that depends on its filtration characteristics. Clearance is a function of [[glomerular filtration]], secretion from the [[peritubular capillaries]] to the [[nephron]], and reabsorption from the [[nephron]] back to the [[peritubular capillaries]].
==Definition==
When referring to the function of the [[kidney]], clearance 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 [[kidney]]s'' 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."<ref>{{cite journal |author=Seldin DW |title=The development of the clearance concept |journal=J. Nephrol. |volume=17 |issue=1 |pages=166–71 |year=2004 |pmid=15151274}} Available at: [http://www.sin-italy.org/jnonline/Vol17n1/166.html http://www.sin-italy.org/jnonline/Vol17n1/166.html]. Accessed on: Sept 2, 2007.</ref> From a [[mass transfer]] perspective<ref>{{cite journal |author=Babb AL, Popovich RP, Christopher TG, Scribner BH |title=The genesis of the square meter-hour hypothesis |journal=Transactions - American Society for Artificial Internal Organs |volume=17 |issue= |pages=81–91 |year=1971 |pmid=5158139}}</ref> and [[physiology|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 [[nephron|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 [[blood plasma|plasma]]) concentration''.
Its definition follows from the [[differential equation]] that describes [[exponential decay]] and is used to model kidney function and [[hemodialysis]] machine function:
<math>V \frac{dC}{dt} = -K \cdot C + \dot{m} \qquad (1)</math>
Where:
*<math>\dot{m}</math> 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 <math>\frac{dC}{dt}</math> 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]]:
:<math>\Delta m_{body}=(-\dot m_{out}+ \dot m_{in} +\dot m_{gen.})\Delta t \qquad (2)</math>
where:
*<math>\Delta t</math> is a period of time
*<math>\Delta m_{body}</math> the change in mass of the toxin in the body during <math>\Delta t</math>
*<math>\dot m_{in}</math> is the toxin intake rate
*<math>\dot m_{out}</math> is the toxin removal rate
*<math>\dot m_{gen.}</math> is the toxin generation rate
In words, the above equation states:
:''The change in the mass of a toxin within the body (<math>\Delta m</math>) during some time <math>\Delta t</math> is equal to the toxin intake plus the toxin generation minus the toxin removal.
Since
:<math>m_{body} = C \cdot V \qquad (3)</math>
and
:<math>\dot m_{out}=K \cdot C \qquad (4)</math>
Equation A1 can be re-written as:
:<math>\Delta (C \cdot V)=(-K \cdot C+ \dot m_{in} +\dot m_{gen.})\Delta t \qquad (5)</math>
If one lumps the ''in'' and ''gen.'' terms together, i.e. <math>\dot m=\dot m_{in} +\dot m_{gen.}</math> and divides by <math>\Delta t</math> the result is a [[difference equation]]:
:<math>\frac{\Delta (C \cdot V)}{\Delta t} = -K \cdot C + \dot{m} \qquad(6)</math>
If one applies the [[limit (mathematics)|limit]] <math>\Delta t \rightarrow 0</math> one obtains a differential equation:
:<math>\frac{d(C \cdot V)}{dt}= -K \cdot C + \dot{m} \qquad(7)</math>
Using the [[chain rule]] this can be re-written as:
:<math>C \frac{dV}{dt}+V \frac{dC}{dt} = -K \cdot C + \dot{m} \qquad(8)</math>
If one assumes that the volume change is not significant, i.e. <math>C \frac{dV}{dt}=0</math>, the result is Equation ''1'':
:<math>V \frac{dC}{dt} = -K \cdot C + \dot{m} \qquad(1)</math>
==Solution to the differential equation==
The general solution of the above differential equation (''1'') is:
<math>C = \frac{\dot{m}}{K} + (C_{o}-\frac{\dot{m}}{K}) e^{-\frac{K \cdot t}{V}} \qquad (9)</math><ref name=gotch1998>{{cite journal |author=Gotch FA |title=The current place of urea kinetic modelling with respect to different dialysis modalities |journal=Nephrol. Dial. Transplant. |volume=13 Suppl 6 |issue= |pages=10–4 |year=1998 |pmid=9719197 |doi=10.1093/ndt/13.suppl_6.10}} [http://ndt.oxfordjournals.org/cgi/reprint/13/suppl_6/10 Full Text]</ref><ref name=gotch2000>{{cite journal |author=Gotch FA, Sargent JA, Keen ML |title=Whither goest Kt/V? |journal=Kidney Int. Suppl. |volume=76 |issue= |pages=S3–18 |year=2000 |pmid=10936795 |doi=10.1046/j.1523-1755.2000.07602.x}}</ref>
Where:
*C<sub>o</sub> 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 (mathematical constant)|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:
<math> C_{\infty} = \frac {\dot{m}}{K} \qquad (10a)</math>
The above equation (''10a'') can be re-written as:
<math> K = \frac {\dot{m}}{C_{\infty}} \qquad (10b)</math>
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 proportional|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'')):
<math>K = \frac {C_U \cdot Q}{C_B} \qquad (11)</math>
Where:
*K is the clearance [mL/min]
*C<sub>U</sub> 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])
*C<sub>B</sub> 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'').
==See also==
*[[Table of medication secreted in kidney]]
*[[Sieving coefficient]]
*[[Creatinine clearance]]
*[[Kt/V]]
*[[Pharmacokinetics]]
*[[Renal clearance ratio]]
*[[Standardized Kt/V]]
*[[Urea reduction ratio]]
{{renal physiology}}
{{Pharmacology}}
==References==
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[[Category:nephrology]]
[[Category:pharmacokinetics]]
[[Category:pharmacology]]
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