Kt/V

K (clearance) multiplied by t (time) is a volume (since mL/min × min = mL, or L/h × h = L), and (K × t) can be thought of as the mL or L of fluid (blood in this case) cleared of urea (or any other solute) during the course of a single treatment. V also is a volume, expressed in mL or L. So the ratio of K × t / V is a so-called "dimensionless ratio" and can be thought of as a multiple of the volume of plasma cleared of urea divided by the distribution volume of urea. When Kt/V = 1.0, a volume of blood equal to the distribution volume of urea has been completely cleared of urea.

The relationship between Kt/V and the concentration of urea C at the end of dialysis can be derived from the first-order differential equation that describes exponential decay and models the clearance of any substance from the body where the concentration of that substance decreases in an exponential fashion:

${\displaystyle V{\frac {dC}{dt}}=-KC}$

(1)

where

• C is the concentration [mol/m³]
• t is the time [s]
• K is the clearance [m³/s]
• V is the volume of distribution [m³]

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.

This equation is separable and can be integrated (assuming K and V are constant) as follows:

${\displaystyle \int {\frac {1}{C}}dC=\int -{\frac {K}{V}}\,dt.}$

(2a)

After integration,

${\displaystyle \ln(C)=-{\frac {Kt}{V}}+{\mbox{c}}}$

(2b)

where

• c is the constant of integration

If one takes the antilog of equation 2b the result is:

${\displaystyle C=e^{-{\frac {Kt}{V}}+\mathrm {c} }}$

(2c)

where

• e is the base of the natural logarithm

By integer exponentiation this can be written as:

${\displaystyle C=C_{0}e^{-{\frac {Kt}{V}}}}$

(3)

where

• C₀ is the concentration at the beginning of dialysis [mmol/L] or [mol/m³].

The above equation can also be written as[2]

${\displaystyle {\frac {Kt}{V}}=\ln {\frac {C_{o}}{C}}}$

(4)

Normally we measure postdialysis serum urea nitrogen concentration C and compare this with the initial or predialysis level C₀. The session length or time is t and this is measured by the clock. The dialyzer clearance K is usually estimated, based on the urea transfer ability of the dialyzer (a function of its size and membrane permeability), the blood flow rate, and the dialysate flow rate.[4] In some dialysis machines, the urea clearance during dialysis is estimated by testing the ability of the dialyzer to remove a small salt load that is added to the dialysate during dialysis.

The URR or Urea reduction ratio is simply the fractional reduction of urea during dialysis. So by definition, URR = 1 − C/C₀. So 1−URR = C/C₀. So by algebra, substituting into equation (4) above, since ln C/C₀ = − ln C₀/C, we get:

${\displaystyle {\frac {Kt}{V}}=-\ln(1-URR).}$

(8)

Kt/V (in the context of peritoneal dialysis) was developed by Michael J. Lysaght in a series of articles on peritoneal dialysis.[5][6]

The steady-state solution of a simplified mass transfer equation that is used to describe the mass exchange over a semi-permeable membrane and models peritoneal dialysis is

${\displaystyle C_{B}={\dot {m}}/K.}$

(6a)

where

• C_B is the concentration in the blood [ mol/m³ ]
• K_D is the clearance [ m³/s ]
• ${\displaystyle {\dot {m}}}$ is the urea mass generation [ mol/s ]

This can also be written as:

${\displaystyle K={\dot {m}}/C_{B}.}$

(6b)

The mass generation (of urea), in steady state, can be expressed as the mass (of urea) in the effluent per time:

${\displaystyle {\dot {m}}={\frac {C_{E}\cdot V_{E}}{t}}.}$

(6c)

where

• C_E is the concentration of urea in effluent [ mol/m³ ]
• V_E is the volume of effluent [ m³ ]
• t is the time [ s ]

Lysaght, motivated by equations 6b and 6c, defined the value K_D:

${\displaystyle K_{D}={\frac {C_{E}\cdot V_{E}}{C_{B}\cdot t}}}$

(6d)

Lysaght uses "ml/min" for the clearance. In order to convert the above clearance (which is in m³/s) to ml/min one has to multiply by 60 × 1000 × 1000.

Once K_D is defined the following equation is used to calculate Kt/V:

${\displaystyle {\frac {K\cdot t}{V}}={\frac {7/3\cdot K_{D}}{V_{D}}}}$

(7a)

where

• V is the volume of distribution. It has to be in litres (L), as the equation is not really non-dimensional.

The 7/3 is used to adjust the Kt/V value so it can be compared to the Kt/V for hemodialysis, which is typically done thrice weekly in the USA.

Kt/V has been widely adopted because it was correlated with survival. Before Kt/V nephrologists measured the serum urea concentration (specifically the time-averaged concentration of urea (TAC of urea)), which was found not to be correlated with survival (due to its strong dependence on protein intake) and thus deemed an unreliable marker of dialysis adequacy.

• It is complex and tedious to calculate. Many nephrologists have difficulty understanding it.
• Urea is not associated with toxicity.[7]
• Kt/V only measures a change in the concentration of urea and implicitly assumes the clearance of urea is comparable to other toxins. (It ignores molecules larger than urea having diffusion-limited transport - so called middle molecules).
• Kt/V does not take into account the role of ultrafiltration.
• It ignores the mass transfer between body compartments and across the plasma membrane (i.e. intracellular to extracellular transport), which has been shown to be important for the clearance of molecules such as phosphate. Practical use of Kt/V requires adjustment for rebound of the urea concentration due to the multi-compartmental nature of the body.
• Kt/V may disadvantage women and smaller patients in terms of the amount of dialysis received. Normal kidney function may be modeled as optimal Glomerular filtration rate or GFR. GFR is usually normalized in people to body surface area. A man and a woman of similar body surface areas will have markedly different levels of total body water (which corresponds to V). Also, smaller people of either sex will have markedly lower levels of V, but only slightly lower levels of body surface area. For this reason, any dialysis dosing system that is based on V may tend to underdose smaller patients and women. Some investigators have proposed dosing based on surface area (S) instead of V, but clinicians usually measure the URR and then calculate Kt/V. One can "adjust" the Kt/V, to calculate a "surface-area-normalized" or "SAN"-Kt/V as well as a "SAN"-standard Kt/V. This puts a wrapper around Kt/V and normalizes it to body surface area.[8]