Standardized Kt/V

Standardized Kt/V, also std Kt/V, is a way of measuring (

• ${\displaystyle {\dot {m}}}$ 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 [min] or [s]
• V is the volume of distribution (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.

Derivation equation 1 is described in the article

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

${\displaystyle C={\frac {\dot {m}}{K}}+\left(C_{o}-{\frac {\dot {m}}{K}}\right)e^{-{\frac {K\cdot t}{V}}}\qquad (2)}$

where

• C_o is the concentration at the beginning of dialysis [mmol/L] or [mol/m³]
• e is the base of the natural logarithm

The steady state solution is

${\displaystyle C_{\infty }={\frac {\dot {m}}{K}}\qquad (3a)}$

This can be written as

${\displaystyle K={\frac {\dot {m}}{C_{\infty }}}\qquad (3b)}$

Equation 3b is the equation that defines

${\displaystyle {K'}={\frac {\dot {m}}{C_{o}}}\qquad (4)}$

where

• K' is the equivalent clearance [mL/min] or [m³/s]
• ${\displaystyle {\dot {m}}}$ is the mass generation rate of the substance - assumed to be a constant, i.e. not a function of time [mmol/min] or [mol/s]
• C_o is the concentration at the beginning of dialysis [mmol/L] or [mol/m³]

Equation 4 is normalized by the volume of distribution to form equation 5:

${\displaystyle {\frac {K'}{V}}={\frac {\dot {m}}{C_{o}\cdot V}}\qquad (5)}$

Equation 5 is multiplied by an arbitrary constant to form equation 6:

${\displaystyle {\mbox{const}}\cdot {\frac {K'}{V}}={\mbox{const}}\cdot {\frac {\dot {m}}{C_{o}\cdot V}}\qquad (6)}$

Equation 6 is then defined as standardized Kt/V (std Kt/V):

${\displaystyle {\mbox{std}}{\frac {K\cdot t}{V}}\ {\stackrel {\mathrm {def} }{=}}\ {\mbox{const}}\cdot {\frac {\dot {m}}{C_{o}\cdot V}}\qquad (7)}$^[1][2]

where

• const is 7×24×60×60 seconds, the number of seconds in a week.

Interpretation of std Kt/V

Standardized Kt/V can be interpreted as a concentration normalized by the mass generation per unit volume of body water.

Equation 7 can be written in the following way:

${\displaystyle {\mbox{std}}{\frac {K\cdot t}{V}}\ {\stackrel {\mathrm {def} }{=}}{\mbox{ const}}\cdot {\frac {\dot {m}}{V}}{\frac {1}{C_{o}}}\qquad (8)}$

If one takes the inverse of Equation 8 it can be observed that the inverse of std Kt/V is proportional to the concentration of urea (in the body) divided by the production of urea per time per unit volume of body water.

${\displaystyle \left[std{\frac {K\cdot t}{V}}\right]^{-1}\propto {\frac {C_{o}}{{\dot {m}}/V}}\qquad (9)}$

Comparison to Kt/V

Kt/V and standardized Kt/V are not the same. Kt/V is a ratio of the pre- and post-dialysis urea concentrations. Standardized Kt/V is an equivalent clearance defined by the initial urea concentration (compare equation 8 and equation 10).

Kt/V is defined as (see article on Kt/V for derivation):

${\displaystyle {\frac {K\cdot t}{V}}=\ln {\frac {C_{o}}{C}}\qquad (10)}$^[3]

Since Kt/V and std Kt/V are defined differently, Kt/V and std Kt/V values cannot be compared.

Advantages of std Kt/V

• Can be used to compare any dialysis schedule (i.e.
  nocturnal home hemodialysis
  vs. daily hemodialysis vs. conventional hemodialysis)
• Applicable to peritoneal dialysis.
• Can be applied to patients with residual renal function; it is possible to demonstrate that C_o is a function of the residual kidney function and the "cleaning" provided by dialysis.
• The model can be applied to substances other than urea, if the clearance, K, and generation rate of the substance, ${\displaystyle {\dot {m}}}$, are known.^[2]

Criticism/disadvantages of std Kt/V

• It is complex and tedious to calculate, although web-based calculators are available to do this fairly easily.
• Many nephrologists have difficulty understanding it.
• Urea is not associated with toxicity.^[4]
• Standardized Kt/V only models the clearance of urea and thus implicitly assumes the clearance of urea is comparable to other toxins. It ignores molecules that (relative to urea) have diffusion-limited transport - so called middle molecules.
• It ignores the
  extracellular transport), which has been shown to be important for the clearance of molecules such as phosphate
  .
• The Standardized Kt/V is based on body water volume (V). The Glomerular filtration rate, an estimate of normal kidney function, is usually normalized to body surface area (S). S and V differ markedly between small vs. large people and between men and women. A man and a woman of the same S will have similar levels of GFR, but their values for V may differ by 15-20%. Because standardized Kt/V incorporates residual renal function into the calculations, it makes the assumption that kidney function should scale by V. This may disadvantage women and smaller patients of either sex, in whom V is decreased to a greater extent than S.

Calculating stdKt/V from treatment Kt/V and number of sessions per week

The various ways of computing standardized Kt/V by Gotch,^[1] Leypoldt,^[5] and the FHN trial network ^[6] are all a bit different, as assumptions differ on equal spacing of treatments, use of a fixed or variable volume model, and whether or not urea rebound is taken into effect.^[7] One equation, proposed by Leypoldt and modified by Depner that is cited in the KDOQI 2006 Hemodialysis Adequacy Guidelines and which is the basis for a web calculator for stdKt/V is as follows:

${\displaystyle stdKt/V={\frac {\frac {10080\cdot (1-e^{-eKt/V})}{t}}{{\frac {1-e^{-eKtV}}{spKt/V}}+{\frac {10080}{N\cdot t}}-1}}}$

where stdKt/V is the standardized Kt/V
spKt/V is the single-pool Kt/V, computed as described in Kt/V section using a simplified equation or ideally, using urea modeling, and
eKt/V is the equilibrated Kt/V, computed from the single-pool Kt/V (spKt/V) and session length (t) using, for example, the Tattersall equation:^[8]

${\displaystyle ekt/V=spKt/V\cdot {\frac {t}{t+C}}}$

where t is session duration in minutes, and C is a time constant, which is specific for type of access and type solute being removed. For urea, C should be 35 minutes for arterial access and 22 min for a venous access.

The regular "rate equation" ^[9] also can be used to determine equilibrated Kt/V from the spKt/V, as long as session length is 120 min or longer.

Plot showing std Kt/V depending on regular Kt/V for different treatment regimens

One can create a plot to relate the three grouping (standardized Kt/V, Kt/V, treatment frequency per week), sufficient to define a dialysis schedule. The equations are strongly dependent on session length; the numbers will change substantially between two sessions given at the same schedule, but with different session lengths.^{[citation needed]}

For the present plot, a session length of 0.4 Kt/V units per hour was assumed, with a minimum dialysis session length of 2.0 hours.

References