**1. Introduction**

The mathematical description of hemodialysis (HD) includes two parts: 1) explanation of the exchange between patient's blood and dialysate fluid across a semipermeable membrane of the dialyzer, and 2) characterization of the solute removal from the patient. The solute transport across the dialyzer membrane depends on the difference in hydrostatic pressure and solute concentration gradients between both sides of the membrane and also on the permeability of the membrane to the solute. The local equations for solute and fluid transport through the membrane are based on a phenomenological (thermodynamic) description according to the Staverman-Kedem-Katchalsky-Spiegler approach (Staverman, 1951; Kedem & Katchalsky, 1958; Katchalsky & Curran, 1965; Spiegler & Kedem, 1966). The two compartment model describes the functioning of the patient – dialyzer system, assuming that body fluid is divided into two parts: one directly (extracellular compartment) and one indirectly (intracellular compartment) accessible for dialysis (Schneditz & Daugirdas, 2001). The one compartment model of the solute distribution volume assumes that the solute is distributed in a single, homogenous pool. Solute kinetic modeling is based on a set of ordinary differential equations describing the changes of solute mass, concentration and distribution volume in body compartments and in the dialyzer. Using solute kinetic modeling one is able to evaluate dialysis efficiency.

The question concerning dialysis dosing has been debated and remains controversial since the beginning of the dialysis treatment era. Between 1976 and 1981, the National Cooperative Dialysis Study (NCDS) was performed in the United States to establish objective, quantitative criteria for the adequate dose of dialysis (Gotch & Sargent, 1985; Sargent & Gotch, 1989; Locatelli et al., 2005). The primary analysis showed that morbidity was less at lower levels of time average urea concentration. The secondary 'mechanistic' analysis of the NCDS data done by Gotch and Sargent launched the issue of urea KT/V (Gotch & Sargent, 1985).

Single-pool KT/V overestimates the removed amount of urea because of the postdialysis urea rebound, i.e., a fast postdialysis increase in urea concentration in plasma, which is a compartmental effect; therefore, the equilibrated KT/V (eqKT/V), estimated by the Daugirdas formula, was introduced to clinical practice (Daugirdas et al., 2001). Equilibrated KT/V values can be also calculated using an alternative equation by Daugirdas and

Kinetic Modeling and Adequacy of Dialysis 5

The mathematical description of hemodialysis includes two parts: 1) one part that explains the fluid and solute transport across a semi-permeable membrane of the dialyzer, and 2) one part that characterizes the global solute transport between removal device and patient.

The fluid and solute transport in dialyzer consists of two processes: transport through a permselective membrane between blood and dialysate and transport in blood and dialysate

The theoretical description of transport through a permselective membrane is based on phenomenological (thermodynamic) descriptions according to the Staverman-Kedem-Katchalsky-Spiegler approach (Staverman, 1951; Kedem & Katchalsky, 1958; Katchalsky & Curran, 1965; Spiegler & Kedem, 1966; Weryński & Nowosielcew, 1983; Werynski & Waniewski, 1995; Waniewski, 2006). Diffusion is the dominant factor for small solute transport in hemodialyzer. The transport due to convection prevails in hemofilters, plasma separators, etc. In hemodialyzer with highly permeable membrane used in hemodiafiltration, the convective transport component plays a leading role in the removal of

Considering the dialyzer as shown in Fig. 1, the system will soon after the start of dialysis be

where Qb,o = Qb,i – Qv and Qd,o = Qd,i + Qv are the rates of blood and dialysate flows at the outlet of hemodialyzer, respectively, Qv is ultrafiltration rate, Cb,i and Cd,i are the inlet blood and dialysate concentrations and Cb,o and Cd,o are the outlet blood and dialysate

Cb,i Cb,o, Qb,o , Qb,i

The left side of equation (2) represents the solute leaving the blood; the right side is the solute appearing in dialysate. The first term on each side of equation (2) is the diffusive

At any specific blood and dialysis fluid flow rates, the diffusive dialysance D is the change

component of flux and the second term represents the convective contribution.

in solute amount of incoming blood over concentration driving force (Cb,i – Cd,i):

QC QC Q Q C Q Q C b,i b,i d,i d,i b,i v b,o d,i v d,o + =− ++ ( ) ( ) (1)

Q C C QC Q C C QC b,i b,i b,o v b,o d,i d,o d,i v d,o ( − + = −+ ) ( ) (2)

Cd,i, Qd,i

**2. Theory of fluid and solute transport in hemodialysis** 

middle molecules and small proteins (Werynski & Waniewski, 1995).

Fig. 1. Schematic description of concentration and flows in dialyzer.

Cd,o, Qd,o

**2.1 Solute and fluid transport in dialyzer** 

at the quasi-steady state with the mass balance:

concentrations, respectively.

After rearrangement of equation (1):

channels.

Schneditz (Daugirdas & Schneditz, 1995), or the formula derived from observations during the HEMO Study (Depner et al., 1999; Eknoyan et al., 2002; Daugirdas et al., 2004), or that introduced by Tattersall et al. (Tattersall et al., 1996).

The usage of the KT/V index as a sole and optimal measure of dialysis dose is questioned by many authors. Fractional solute removal (FSR) and equivalent continuous clearance (ECC) are two such alternative options, which can be used instead of KT/V. FSR was suggested by Verrina et al. (Verrina et al., 1998) and Henderson (Henderson, 1999) for comparative studies of various dialysis modalities and schedules. By definition FSR is the removed mass over the reference solute mass in the body. The concept of FSR is closely related to the concept of the solute removal index (SRI) proposed by Keshaviah (Keshaviah, 1995). Standard KT/V (stdKT/V), introduced by Gotch, is another variant of FSR (Gotch, 1998). The time-average solute concentration (Cta) has been introduced to define 'equivalent renal clearance' (EKR), as a solute removal rate over Cta (Casino & Lopez, 1996). Using other reference concentrations in the definition of EKR instead of Cta, the general idea of equivalent continuous clearance, ECC, can be formulated (Waniewski et al., 2006; Waniewski et al., 2010). There are at least four different reference methods: 1) peak, p, 2) peak average, pa, 3) time average, ta, and 4) treatment time average, trta, reference values of volume, mass, and concentration applied in KT/V, FSR and ECC (Waniewski et al., 2006; Waniewski et al., 2010). KT/V, FSR and ECC are mathematically related for the same reference method. However, the choice of an adequacy index and the respective reference method is not obvious. It is not possible to decide whether this or the other definition is better although some authors have declared their preferences (Keshaviah, 1995; Casino & Lopez, 1996; Verrina et al., 1998; Henderson, 1999). The difference between different hypotheses and the indices based on them may be investigated theoretically, but the choice, if any, may be done only on the basis of a large set of clinical data. Future research should hopefully provide more information about the relationship between various definitions and the probability of clinical outcome in dialyzed patients.

Recent studies report some advantages of low-efficiency, frequent schedule over short, highefficiency HD (Depner, 1998; Charra et al., 2004). The two compartment variable volume urea kinetic model can be applied to examine the whole set of dialysis adequacy indices in different dialysis treatments, e.g. 1) conventional HD with 3 sessions per week, 2) daily HD with 6 sessions per week and 3) nocturnal HD with 6 long sessions using typical patient and treatment parameters. The peak average reference method used in FSR and ECC calculations seem to be a more sensitive to the frequency and time of dialysis than the method based on time average reference (Waniewski et al., 2006; Waniewski et al., 2010).

The unified approach to the definition of dialysis adequacy indices proposed by Waniewski et al. is valid for all modalities of dialysis performed in end-stage renal disease and acute renal failure patients and for the assessment of residual renal function (Waniewski et al., 2006; Debowska et al., 2010; Waniewski et al., 2010). The integrated system of dialysis adequacy indices takes into account all currently applied indices and allows to explain their relationships and specificities.

The theory and practical application of this system of adequacy indices are here presented on the basis of our previous publications and a (unpublished) PhD thesis (Waniewski & Lindholm, 2004; Debowska & Waniewski, 2005; Debowska et al., 2005; Waniewski et al., 2006; Debowska et al., 2007a; Debowska et al., 2007b; Debowska et al., 2010; Waniewski et al., 2010).

Schneditz (Daugirdas & Schneditz, 1995), or the formula derived from observations during the HEMO Study (Depner et al., 1999; Eknoyan et al., 2002; Daugirdas et al., 2004), or that

The usage of the KT/V index as a sole and optimal measure of dialysis dose is questioned by many authors. Fractional solute removal (FSR) and equivalent continuous clearance (ECC) are two such alternative options, which can be used instead of KT/V. FSR was suggested by Verrina et al. (Verrina et al., 1998) and Henderson (Henderson, 1999) for comparative studies of various dialysis modalities and schedules. By definition FSR is the removed mass over the reference solute mass in the body. The concept of FSR is closely related to the concept of the solute removal index (SRI) proposed by Keshaviah (Keshaviah, 1995). Standard KT/V (stdKT/V), introduced by Gotch, is another variant of FSR (Gotch, 1998). The time-average solute concentration (Cta) has been introduced to define 'equivalent renal clearance' (EKR), as a solute removal rate over Cta (Casino & Lopez, 1996). Using other reference concentrations in the definition of EKR instead of Cta, the general idea of equivalent continuous clearance, ECC, can be formulated (Waniewski et al., 2006; Waniewski et al., 2010). There are at least four different reference methods: 1) peak, p, 2) peak average, pa, 3) time average, ta, and 4) treatment time average, trta, reference values of volume, mass, and concentration applied in KT/V, FSR and ECC (Waniewski et al., 2006; Waniewski et al., 2010). KT/V, FSR and ECC are mathematically related for the same reference method. However, the choice of an adequacy index and the respective reference method is not obvious. It is not possible to decide whether this or the other definition is better although some authors have declared their preferences (Keshaviah, 1995; Casino & Lopez, 1996; Verrina et al., 1998; Henderson, 1999). The difference between different hypotheses and the indices based on them may be investigated theoretically, but the choice, if any, may be done only on the basis of a large set of clinical data. Future research should hopefully provide more information about the relationship between various definitions and

Recent studies report some advantages of low-efficiency, frequent schedule over short, highefficiency HD (Depner, 1998; Charra et al., 2004). The two compartment variable volume urea kinetic model can be applied to examine the whole set of dialysis adequacy indices in different dialysis treatments, e.g. 1) conventional HD with 3 sessions per week, 2) daily HD with 6 sessions per week and 3) nocturnal HD with 6 long sessions using typical patient and treatment parameters. The peak average reference method used in FSR and ECC calculations seem to be a more sensitive to the frequency and time of dialysis than the method based on time average reference (Waniewski et al., 2006; Waniewski et al., 2010). The unified approach to the definition of dialysis adequacy indices proposed by Waniewski et al. is valid for all modalities of dialysis performed in end-stage renal disease and acute renal failure patients and for the assessment of residual renal function (Waniewski et al., 2006; Debowska et al., 2010; Waniewski et al., 2010). The integrated system of dialysis adequacy indices takes into account all currently applied indices and allows to explain their

The theory and practical application of this system of adequacy indices are here presented on the basis of our previous publications and a (unpublished) PhD thesis (Waniewski & Lindholm, 2004; Debowska & Waniewski, 2005; Debowska et al., 2005; Waniewski et al., 2006; Debowska et al., 2007a; Debowska et al., 2007b; Debowska et al., 2010; Waniewski et

introduced by Tattersall et al. (Tattersall et al., 1996).

the probability of clinical outcome in dialyzed patients.

relationships and specificities.

al., 2010).

### **2. Theory of fluid and solute transport in hemodialysis**

The mathematical description of hemodialysis includes two parts: 1) one part that explains the fluid and solute transport across a semi-permeable membrane of the dialyzer, and 2) one part that characterizes the global solute transport between removal device and patient.

### **2.1 Solute and fluid transport in dialyzer**

The fluid and solute transport in dialyzer consists of two processes: transport through a permselective membrane between blood and dialysate and transport in blood and dialysate channels.

The theoretical description of transport through a permselective membrane is based on phenomenological (thermodynamic) descriptions according to the Staverman-Kedem-Katchalsky-Spiegler approach (Staverman, 1951; Kedem & Katchalsky, 1958; Katchalsky & Curran, 1965; Spiegler & Kedem, 1966; Weryński & Nowosielcew, 1983; Werynski & Waniewski, 1995; Waniewski, 2006). Diffusion is the dominant factor for small solute transport in hemodialyzer. The transport due to convection prevails in hemofilters, plasma separators, etc. In hemodialyzer with highly permeable membrane used in hemodiafiltration, the convective transport component plays a leading role in the removal of middle molecules and small proteins (Werynski & Waniewski, 1995).

Considering the dialyzer as shown in Fig. 1, the system will soon after the start of dialysis be at the quasi-steady state with the mass balance:

$$\mathbf{Q}\_{\rm b,i}\mathbf{C}\_{\rm b,i} + \mathbf{Q}\_{\rm d,i}\mathbf{C}\_{\rm d,i} = \left(\mathbf{Q}\_{\rm b,i} - \mathbf{Q}\_{\rm v}\right)\mathbf{C}\_{\rm b,o} + \left(\mathbf{Q}\_{\rm d,i} + \mathbf{Q}\_{\rm v}\right)\mathbf{C}\_{\rm d,o} \tag{1}$$

where Qb,o = Qb,i – Qv and Qd,o = Qd,i + Qv are the rates of blood and dialysate flows at the outlet of hemodialyzer, respectively, Qv is ultrafiltration rate, Cb,i and Cd,i are the inlet blood and dialysate concentrations and Cb,o and Cd,o are the outlet blood and dialysate concentrations, respectively.

Fig. 1. Schematic description of concentration and flows in dialyzer.

After rearrangement of equation (1):

$$\mathbf{Q}\_{\rm b,l} \left( \mathbf{C}\_{\rm b,l} - \mathbf{C}\_{\rm b,o} \right) + \mathbf{Q}\_{\rm v} \mathbf{C}\_{\rm b,o} = \mathbf{Q}\_{\rm d,l} \left( \mathbf{C}\_{\rm d,o} - \mathbf{C}\_{\rm d,l} \right) + \mathbf{Q}\_{\rm v} \mathbf{C}\_{\rm d,o} \tag{2}$$

The left side of equation (2) represents the solute leaving the blood; the right side is the solute appearing in dialysate. The first term on each side of equation (2) is the diffusive component of flux and the second term represents the convective contribution.

At any specific blood and dialysis fluid flow rates, the diffusive dialysance D is the change in solute amount of incoming blood over concentration driving force (Cb,i – Cd,i):

$$\mathbf{D} = \frac{\mathbf{Q}\_{\mathrm{b},i} \left( \mathbf{C}\_{\mathrm{b},i} - \mathbf{C}\_{\mathrm{b},o} \right)}{\mathbf{C}\_{\mathrm{b},i} - \mathbf{C}\_{\mathrm{d},i}} = \frac{\mathbf{Q}\_{\mathrm{d},i} \left( \mathbf{C}\_{\mathrm{d},o} - \mathbf{C}\_{\mathrm{d},i} \right)}{\mathbf{C}\_{\mathrm{b},i} - \mathbf{C}\_{\mathrm{d},i}} \tag{3}$$

Assuming that solute concentration in the inflowing dialysate is zero (Cd,i = 0) equation (3) yields the definition of diffusive clearance K:

$$\mathbf{K} = \frac{\mathbf{Q}\_{\rm b,l} \left( \mathbf{C}\_{\rm b,l} - \mathbf{C}\_{\rm b,o} \right)}{\mathbf{C}\_{\rm b,l}} \tag{4}$$

Kinetic Modeling and Adequacy of Dialysis 7

K, Qv Kc

Extracellular compartment Me, Ce, Ve

Fig. 2. One and two compartment models for the distribution of water and solutes in the body. solutes (as urea and creatinine) and proteins (as β2-microglobulin). In some papers, extracellular and intracellular water were called perfused and non-perfused compartments,

Kr, Krw

In one compartment model the rate of the change of solute mass in the body, dMb/dt = d(CbVb)/dt, and in dialysate, dMd/dt = d(CdVd)/dt, during hemodialysis, are

( ) ( )

dCV G K C C KC

In the two compartment model, the removal of solute by the dialyzer with clearance K and by the kidneys with residual clearance Kr, is a function of the solute concentration in the extracellular compartment, Ce, but indirectly depends also on the intercompartmental mass

b d

ci e e d r e

b d rb

(7)

Dialyzer Md, Cd, Vd

(8)

<sup>⎪</sup> =− − − <sup>⎪</sup>

( ) ( )

( ) ( )( )

ci e

e d

⎪ = − − − +−

d VC K C C K C C G KC

For urea and creatinine, Cd = 0 in standard hemodialysis and hemofiltration treatments, because fresh dialysis fluid without these solutes is continuously provided. The rate of total

= −+ ( ) <sup>R</sup>

dM K C C KC

e d re

dt (9)

dCV KC C

<sup>⎪</sup> = − ⎪⎩

respectively (Clark et al., 1999; Leypoldt et al., 2003; Leypoldt et al., 2004).

b b

dt

dt

d d

( ) ( )

d VC KC C

⎨ =− −

e e

dt

dt

dt

i i

d d

⎪ = −

( ) ( )

solute mass removal from the body, dMR/dt, during hemodialysis is:

d VC KC C

described by the following ordinary differential equations:

Patient body G, Gw

Mb, Cb, Vb

Intracellular compartment Mi, Ci, Vi

⎧

⎨

⎧

⎪ ⎪

⎪ ⎪

⎩

transport coefficient Kc:

Dialyzer clearance is a parameter that describes the efficiency of membrane devices, i.e. the solute removal rate from the blood related to blood solute concentration at the inlet to the hemodialyzer (Darowski et al., 2000; Waniewski, 2006).

Ultrafiltration Qv from blood to dialysate increases diffusive solute transport from blood to dialysate and therefore the clearance of the hemodialyzer or hemofilter may be described as:

$$\mathbf{K} = \mathbf{K}\_0 + \mathbf{T}\_\mathbf{r} \cdot \mathbf{Q}\_\mathbf{v} \tag{5}$$

where K0 is the diffusive clearance for Qv = 0 and Tr is the transmittance coefficient (Werynski & Waniewski, 1995; Darowski et al., 2000; Waniewski, 2006). Although the dependence of K on Qv in the one-dimensional theory is slightly nonlinear, one may assume the linear description used in equation (5) that was confirmed experimentally with high accuracy (Waniewski et al., 1991). Tr may be estimated from the experimental data using the equation:

$$\mathbf{T}\_r = \frac{\mathbf{K} - \mathbf{K}\_0}{\mathbf{Q}\_v} \tag{6}$$

The measurements of K0 and K for a few different values of Qv allow determining Tr using equation (6) and linear regression.

### **2.2 One and two compartment models for the distribution of fluid and solutes in the body**

Compartment models consider the patient body as a single compartment (thick line in Fig. 2) or as two compartments: intracellular and extracellular (dashed line in Fig. 2).

The one compartment model of the solute distribution volume assumes that solute mass, Mb, is distributed in the body in a single, homogenous pool of volume Vb with concentration Cb. The two compartment model assumes that body fluid is divided into two parts: one directly (extracellular compartment, described by solute mass Me, concentration Ce and fluid volume Ve) and one indirectly (intracellular compartment, with solute mass Mi, concentration Ci and fluid volume Vi) accessible for dialysis (Schneditz & Daugirdas, 2001). It is assumed that solute generation, at the rate G, and water intake, at the rate Gw, occur only in the extracellular space. In the two compartment model, solute and water removal by the kidneys, with clearances Kr and Krw, respectively, are also related only to the extracellular compartment.

Some authors use more general terminology for the two compartment model with perfused and non-perfused compartments, without deciding a priori about their physiological interpretation. This terminology may be used for the description of the distribution of small

Assuming that solute concentration in the inflowing dialysate is zero (Cd,i = 0) equation (3)

Dialyzer clearance is a parameter that describes the efficiency of membrane devices, i.e. the solute removal rate from the blood related to blood solute concentration at the inlet to the

Ultrafiltration Qv from blood to dialysate increases diffusive solute transport from blood to dialysate and therefore the clearance of the hemodialyzer or hemofilter may be described as:

where K0 is the diffusive clearance for Qv = 0 and Tr is the transmittance coefficient (Werynski & Waniewski, 1995; Darowski et al., 2000; Waniewski, 2006). Although the dependence of K on Qv in the one-dimensional theory is slightly nonlinear, one may assume the linear description used in equation (5) that was confirmed experimentally with high accuracy (Waniewski et al., 1991). Tr may be estimated from the experimental data using the

r

K K <sup>T</sup> Q

The measurements of K0 and K for a few different values of Qv allow determining Tr using

**2.2 One and two compartment models for the distribution of fluid and solutes in the** 

2) or as two compartments: intracellular and extracellular (dashed line in Fig. 2).

Compartment models consider the patient body as a single compartment (thick line in Fig.

The one compartment model of the solute distribution volume assumes that solute mass, Mb, is distributed in the body in a single, homogenous pool of volume Vb with concentration Cb. The two compartment model assumes that body fluid is divided into two parts: one directly (extracellular compartment, described by solute mass Me, concentration Ce and fluid volume Ve) and one indirectly (intracellular compartment, with solute mass Mi, concentration Ci and fluid volume Vi) accessible for dialysis (Schneditz & Daugirdas, 2001). It is assumed that solute generation, at the rate G, and water intake, at the rate Gw, occur only in the extracellular space. In the two compartment model, solute and water removal by the kidneys, with clearances Kr and Krw, respectively, are also related only to the

Some authors use more general terminology for the two compartment model with perfused and non-perfused compartments, without deciding a priori about their physiological interpretation. This terminology may be used for the description of the distribution of small

0

v

QC C QC C <sup>D</sup> CC CC

> QC C <sup>K</sup> C

yields the definition of diffusive clearance K:

equation:

**body** 

equation (6) and linear regression.

extracellular compartment.

hemodialyzer (Darowski et al., 2000; Waniewski, 2006).

b,i b,i b,o d,i d,o d,i ( ) ( ) b,i d,i b,i d,i

> b,i b,i b,o ( ) b,i

− − <sup>=</sup> <sup>=</sup> − − (3)

<sup>−</sup> <sup>=</sup> (4)

K K TQ = 0 rv + ⋅ (5)

<sup>−</sup> <sup>=</sup> (6)

Fig. 2. One and two compartment models for the distribution of water and solutes in the body.

solutes (as urea and creatinine) and proteins (as β2-microglobulin). In some papers, extracellular and intracellular water were called perfused and non-perfused compartments, respectively (Clark et al., 1999; Leypoldt et al., 2003; Leypoldt et al., 2004).

In one compartment model the rate of the change of solute mass in the body, dMb/dt = d(CbVb)/dt, and in dialysate, dMd/dt = d(CdVd)/dt, during hemodialysis, are described by the following ordinary differential equations:

$$\begin{cases} \frac{d\left(\mathbf{C}\_{\mathrm{b}}\mathbf{V}\_{\mathrm{b}}\right)}{d\mathbf{t}} = \mathbf{G} - \mathbf{K}\left(\mathbf{C}\_{\mathrm{b}} - \mathbf{C}\_{\mathrm{d}}\right) - \mathbf{K}\_{\mathrm{r}}\mathbf{C}\_{\mathrm{b}}\\ \frac{d\left(\mathbf{C}\_{\mathrm{d}}\mathbf{V}\_{\mathrm{d}}\right)}{d\mathbf{t}} = \mathbf{K}\left(\mathbf{C}\_{\mathrm{b}} - \mathbf{C}\_{\mathrm{d}}\right) \end{cases} \tag{7}$$

In the two compartment model, the removal of solute by the dialyzer with clearance K and by the kidneys with residual clearance Kr, is a function of the solute concentration in the extracellular compartment, Ce, but indirectly depends also on the intercompartmental mass transport coefficient Kc:

$$\begin{cases} \frac{\mathbf{d}\left(\mathbf{V}\_{\mathbf{c}}\mathbf{C}\_{\circ}\right)}{\mathbf{d}\mathbf{t}} = \mathbf{K}\_{\mathrm{c}}\left(\mathbf{C}\_{\mathrm{i}} - \mathbf{C}\_{\circ}\right) - \mathbf{K}\left(\mathbf{C}\_{\circ} - \mathbf{C}\_{\mathrm{d}}\right) + \mathbf{G} - \mathbf{K}\_{\mathrm{r}}\mathbf{C}\_{\circ} \\ \frac{\mathbf{d}\left(\mathbf{V}\_{\mathrm{i}}\mathbf{C}\_{\mathrm{i}}\right)}{\mathbf{d}\mathbf{t}} = -\mathbf{K}\_{\mathrm{c}}\left(\mathbf{C}\_{\mathrm{i}} - \mathbf{C}\_{\circ}\right) \\ \frac{\mathbf{d}\left(\mathbf{V}\_{\mathrm{d}}\mathbf{C}\_{\mathrm{d}}\right)}{\mathbf{d}\mathbf{t}} = \mathbf{K}\left(\mathbf{C}\_{\mathrm{s}} - \mathbf{C}\_{\mathrm{d}}\right) \end{cases} \tag{8}$$

For urea and creatinine, Cd = 0 in standard hemodialysis and hemofiltration treatments, because fresh dialysis fluid without these solutes is continuously provided. The rate of total solute mass removal from the body, dMR/dt, during hemodialysis is:

$$\frac{d\mathbf{M}\_{\text{R}}}{dt} = \mathbf{K} \left( \mathbf{C}\_{\text{o}} - \mathbf{C}\_{\text{d}} \right) + \mathbf{K}\_{\text{r}} \mathbf{C}\_{\text{o}} \tag{9}$$

Kinetic Modeling and Adequacy of Dialysis 9

Cpost and Cpre are postdialysis and predialysis blood urea concentration. KT/V was

KT 0.49pcr 0.16) ln 1 V C ⎛ ⎞ <sup>−</sup> −= − ⎜ ⎟

The primary analysis showed that morbidity was less at lower levels of urea Cta and the number of deaths in patients assigned to groups II and IV was very high (Parker et al., 1983). No significant effect of treatment time was found, although there was a clear trend towards

The 'mechanistic' analysis of the NCDS data done by Gotch and Sargent launched the issue of urea KT/V (Gotch & Sargent, 1985). The patient groups II and IV, with high BUN, had low KT/V values at all levels of pcr and the groups I and III, with low pcr, had low levels of BUN and KT/V. For Kt/V > 0.8 the data base was comprised almost entirely of patient groups I and III with pcr > 0.8. KT/V < 0.8 provided inadequate dialysis with high

The factor KT/V was described as the "fractional clearance of urea" (Gotch & Sargent, 1985). If K is the urea clearance and T is time, the term KT is a volume. The ratio of KT to V expresses the fraction of the urea distribution volume that is totally cleared from urea.

**3.1 Fast hemodialysis: two compartment effects, single-pool and equilibrated KT/V**  The human body has a large number of physical compartments. The mathematical description of body is usually simplified by considering it as single pool (one compartment) or as a few interconnected pools. In a multicompartment model, the solute and fluid

The one compartment model assumes that the body acts as a single, well mixed space and is characterized by: 1) high permeability of cells to the solute being modeled, 2) rapidly flowing blood that transports the solute throughout a totally perfused body. The assumptions of one compartment model for urea or creatinine during dialysis are valid as long as the flux of solute into and out of cells is faster than the flux of solute from the extracellular space accessible to dialysis. When the intercompartment flow between body compartments is too slow and constrained in comparison with the solute removal rate from the perfused compartment, then

With the available high efficiency dialyzers and the tendency to short-time, rapid dialysis at least the two compartment modeling appears to be necessary. The two compartment model assumes solute generation to and removal from the perfused space, which is for urea and creatinine typically the extracellular compartment. This assumption is considered reasonable because urea is produced in the liver and enters body water from the systemic circulation (Sargent & Gotch, 1989). Regarding creatinine, in most studies the previously determined urea distribution volumes for each patient were successfully used as an approximation for creatinine distribution space (Canaud et al., 1995; Clark et al., 1998;

The perfused (extracellular) compartment communicates with the non-perfused compartment (intracellular) according to the concentration gradient with an intercompartmental mass transport coefficient (Kc, mL/min). For a low value of Kc, the

the solute behavior increasingly deviates from that of one compartment kinetics.

pre

(14)

⎝ ⎠

prescribed in the NCDS as a function of pcr and Cpre:

a benefit from longer dialysis (p = 0.06).

probability of failure irrespective of pcr.

Waikar & Bonventre, 2009).

transport between body spaces should be described.

The total solute amount removed from the body ΔMR is the mass removed by dialyzer with clearance K and by the kidneys with residual clearance Kr. The solute removal by dialyzer is proportional to the solute concentration gradient between dialysate and extracellular compartment (Ce – Cd) when using the two compartment model. In the one compartment model, the body solute concentration Cb is used in equation (9) instead of Ce.

In the two compartment model, the changes of fluid volume in extracellular and intracellular compartments, Ve(t) and Vi(t), respectively, are assumed to be proportional to the volumes of these compartments (Canaud et al., 1995; Clark et al., 1998; Ziolko et al., 2000):

$$\mathbf{V}\_{\circ}(\mathbf{t}) = \mathbf{a} \cdot \mathbf{V}\_{\circ}(\mathbf{t}), \text{ V}\_{\circ}(\mathbf{t}) = (1 - \mathbf{a}) \cdot \mathbf{V}\_{\circ}(\mathbf{t}) \tag{10}$$

where α is usually about 1/3, Vb for urea and creatinine is assumed to be equal to total body water (TBW) and Vb as well as Ve can be measured by bioimpedance (Zaluska et al., 2002). During HD the change of solute distribution volume is described by a linear relationship:

$$\mathbf{V}\_{\rm b}(\mathbf{t}) = \mathbf{V}\_{\rm b}(\mathbf{t}\_{\rm o}) + \boldsymbol{\beta} \cdot \mathbf{t} \tag{11}$$

where Vb(t0) is the initial volume of solute distribution and the rate of volume change:

$$
\beta = \mathbf{G}\_{\rm w} - \mathbf{K}\_{\rm rw} - \mathbf{Q}\_{\rm v} \tag{12}
$$

consists of water intake with rate Gw, residual water clearance Krw and ultrafiltration with rate Qv.

### **3. Hemodialysis efficiency: history and definitions of dialysis adequacy indices**

The questions concerning how to quantify dialysis dose and how much dialysis should be provided, are controversial and have been debated since the beginning of the dialysis treatment era. Between 1976 and 1981, the National Cooperative Dialysis Study (NCDS) was performed in the United States to establish objective, quantitative criteria for the adequate dose of dialysis (Gotch & Sargent, 1985; Sargent & Gotch, 1989; Locatelli et al., 2005). It included 165 patients and had a 2 x 2 factorial design: the patients were randomized to two different midweek pre-dialysis blood urea nitrogen (BUN) levels (70 vs. 120 mg/dL) and two different treatment times (2.5 - 3.5 vs. 4.5 – 5.0 h).

Concentration targeting in this study used a time average BUN concentration (Cta) of 50 mg/dL (groups I and III) and 100 mg/dL (groups II and IV). Dialysis time was fixed for the protocol; hence, dialyzer clearance was the main treatment parameter that was adjusted. A one compartment variable volume model was used to prescribe and control the treatment. Urea kinetic modeling was applied to determine protein catabolic rate (pcr) and the parameters of dialysis necessary to achieve a specified BUN level with thrice weekly treatments. BUN changes in an individual patient were quantified as the product of dialyzer urea clearance (K, mL/min) and the treatment time (T, min), normalized to the urea distribution volume (V, mL). KT/V exponentially determines the total decrease in BUN during a dialysis treatment:

$$\mathbf{C}\_{\rm pot} = \mathbf{C}\_{\rm pro} \mathbf{e}^{\frac{\mathbf{kT}}{\mathbf{V}}} \tag{13}$$

The total solute amount removed from the body ΔMR is the mass removed by dialyzer with clearance K and by the kidneys with residual clearance Kr. The solute removal by dialyzer is proportional to the solute concentration gradient between dialysate and extracellular compartment (Ce – Cd) when using the two compartment model. In the one compartment

In the two compartment model, the changes of fluid volume in extracellular and intracellular compartments, Ve(t) and Vi(t), respectively, are assumed to be proportional to the volumes of these compartments (Canaud et al., 1995; Clark et al., 1998; Ziolko et al.,

where α is usually about 1/3, Vb for urea and creatinine is assumed to be equal to total body water (TBW) and Vb as well as Ve can be measured by bioimpedance (Zaluska et al., 2002). During HD the change of solute distribution volume is described by a linear relationship:

consists of water intake with rate Gw, residual water clearance Krw and ultrafiltration with

The questions concerning how to quantify dialysis dose and how much dialysis should be provided, are controversial and have been debated since the beginning of the dialysis treatment era. Between 1976 and 1981, the National Cooperative Dialysis Study (NCDS) was performed in the United States to establish objective, quantitative criteria for the adequate dose of dialysis (Gotch & Sargent, 1985; Sargent & Gotch, 1989; Locatelli et al., 2005). It included 165 patients and had a 2 x 2 factorial design: the patients were randomized to two different midweek pre-dialysis blood urea nitrogen (BUN) levels (70 vs. 120 mg/dL) and

Concentration targeting in this study used a time average BUN concentration (Cta) of 50 mg/dL (groups I and III) and 100 mg/dL (groups II and IV). Dialysis time was fixed for the protocol; hence, dialyzer clearance was the main treatment parameter that was adjusted. A one compartment variable volume model was used to prescribe and control the treatment. Urea kinetic modeling was applied to determine protein catabolic rate (pcr) and the parameters of dialysis necessary to achieve a specified BUN level with thrice weekly treatments. BUN changes in an individual patient were quantified as the product of dialyzer urea clearance (K, mL/min) and the treatment time (T, min), normalized to the urea distribution volume (V, mL).

KT/V exponentially determines the total decrease in BUN during a dialysis treatment:

<sup>V</sup> C Ce post pre

KT

= (13)

−

where Vb(t0) is the initial volume of solute distribution and the rate of volume change:

**3. Hemodialysis efficiency: history and definitions of dialysis adequacy** 

two different treatment times (2.5 - 3.5 vs. 4.5 – 5.0 h).

V (t) e bi =⋅ = − ⋅ α V (t), V (t) 1( α) V (t) b (10)

V (t) V t b b0 = ( ) + ⋅ β t (11)

β =− − GK Q w rw v (12)

model, the body solute concentration Cb is used in equation (9) instead of Ce.

2000):

rate Qv.

**indices** 

Cpost and Cpre are postdialysis and predialysis blood urea concentration. KT/V was prescribed in the NCDS as a function of pcr and Cpre:

$$-\frac{\text{KT}}{\text{V}} = \ln\left(1 - \frac{0.49 \text{pcr} - 0.16}{\text{C}\_{\text{pre}}}\right) \tag{14}$$

The primary analysis showed that morbidity was less at lower levels of urea Cta and the number of deaths in patients assigned to groups II and IV was very high (Parker et al., 1983). No significant effect of treatment time was found, although there was a clear trend towards a benefit from longer dialysis (p = 0.06).

The 'mechanistic' analysis of the NCDS data done by Gotch and Sargent launched the issue of urea KT/V (Gotch & Sargent, 1985). The patient groups II and IV, with high BUN, had low KT/V values at all levels of pcr and the groups I and III, with low pcr, had low levels of BUN and KT/V. For Kt/V > 0.8 the data base was comprised almost entirely of patient groups I and III with pcr > 0.8. KT/V < 0.8 provided inadequate dialysis with high probability of failure irrespective of pcr.

The factor KT/V was described as the "fractional clearance of urea" (Gotch & Sargent, 1985). If K is the urea clearance and T is time, the term KT is a volume. The ratio of KT to V expresses the fraction of the urea distribution volume that is totally cleared from urea.

#### **3.1 Fast hemodialysis: two compartment effects, single-pool and equilibrated KT/V**

The human body has a large number of physical compartments. The mathematical description of body is usually simplified by considering it as single pool (one compartment) or as a few interconnected pools. In a multicompartment model, the solute and fluid transport between body spaces should be described.

The one compartment model assumes that the body acts as a single, well mixed space and is characterized by: 1) high permeability of cells to the solute being modeled, 2) rapidly flowing blood that transports the solute throughout a totally perfused body. The assumptions of one compartment model for urea or creatinine during dialysis are valid as long as the flux of solute into and out of cells is faster than the flux of solute from the extracellular space accessible to dialysis. When the intercompartment flow between body compartments is too slow and constrained in comparison with the solute removal rate from the perfused compartment, then the solute behavior increasingly deviates from that of one compartment kinetics.

With the available high efficiency dialyzers and the tendency to short-time, rapid dialysis at least the two compartment modeling appears to be necessary. The two compartment model assumes solute generation to and removal from the perfused space, which is for urea and creatinine typically the extracellular compartment. This assumption is considered reasonable because urea is produced in the liver and enters body water from the systemic circulation (Sargent & Gotch, 1989). Regarding creatinine, in most studies the previously determined urea distribution volumes for each patient were successfully used as an approximation for creatinine distribution space (Canaud et al., 1995; Clark et al., 1998; Waikar & Bonventre, 2009).

The perfused (extracellular) compartment communicates with the non-perfused compartment (intracellular) according to the concentration gradient with an intercompartmental mass transport coefficient (Kc, mL/min). For a low value of Kc, the discrepancy between one and two compartment modeling is larger because the immediate intercompartmental flow is precluded (Debowska et al., 2007b).

Assuming one compartment model, a fixed distribution volume (no ultrafiltration) and no generation during the dialysis, as during a short HD session, the concentration of any solute can be described by the equation (Sargent & Gotch, 1989; Daugirdas et al., 2001):

$$\mathbf{C}\_{t} = \mathbf{C}\_{\text{pro}} \cdot \mathbf{e}^{-\mathbf{K} \cdot t/V} \tag{15}$$

Kinetic Modeling and Adequacy of Dialysis 11

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>9</sup> <sup>10</sup> <sup>11</sup> 0.2**

**Time, h**

Fig. 3. The phenomena of the intradialytic drop in urea concentration in plasma (inbound),

C(T ) <sup>R</sup>

C(Teq) is the urea concentration 30 to 60 minutes after the dialysis session. The eqKT/V is typically about 0.2 KT/V unit lower than the spKt/V, but this difference depends on the efficiency, or rate of dialysis (Daugirdas et al., 2001). Equilibrated KT/V values can be also calculated using an alternative equation, as described by Daugirdas and Schneditz

spKT /V eqKT /V spKT /V 0.6 0.03

or the formula derived from observations during the HEMO Study (Depner et al., 1999;

<sup>T</sup> eqKT /V spKT /V T 36 = ⋅ <sup>+</sup>

where T indicates treatment time in minutes. Equations (22) and (23), were derived from regression using the rebounded BUN measured 30 or 60 minutes after dialysis. The Tattersall equation was derived from theoretical considerations of disequilibrium and

T

= −⋅ + (22)

(24)

spKT /V eqKT /V spKT /V 0.39 <sup>T</sup> = −⋅ (23)

e q

eq

<sup>C</sup> <sup>=</sup> (21)

0

**rebound**

**Cpost**

and the postdialysis increase in urea concentration in plasma (rebound).

**0.4**

(Daugirdas & Schneditz, 1995):

Eknoyan et al., 2002; Daugirdas et al., 2004):

or that introduced by Tattersall et al. (Tattersall et al., 1996):

rebound, but the coefficient was derived from fitting to clinical data.

**0.6**

**0.8**

**Urea concentration, mg/mL**

where

**1**

**C pre**

**inbound**

**1.2**

where Ct is the blood concentration of the solute at any time t during dialysis, Cpre is the blood concentration at the beginning of HD, K is the clearance of applied dialyzer, and V is the solute distribution volume.

The single pool KT/V (spKT/V) for urea is determined from equation (15) as the natural logarithm (ln) of the ratio of postdialysis (Cpost) to predialysis (Cpre) plasma urea concentrations (Gotch & Sargent, 1985; Daugirdas et al., 2001):

$$\text{spKT} / \text{V} = -\ln\left(\frac{\mathbf{C}\_{\text{post}}}{\mathbf{C}\_{\text{pw}}}\right) \tag{16}$$

The expression 1 – Cpost/Cpre, is called urea reduction ratio (URR):

$$\text{URR} = \mathbf{1} - \mathbf{R} \tag{17}$$

where

$$\mathbf{R} = \frac{\mathbf{C}\_{\text{post}}}{\mathbf{C}\_{\text{pw}}} \tag{18}$$

A solute like urea or creatinine is however removed during hemodialysis more efficiently from the extracellular than from the intracellular compartment and its concentration in plasma falls faster than expected when assessed by one compartment modeling; this effect is called urea inbound (Daugirdas et al., 2001), Fig. 3. When dialysis is completed, the flow from intracellular to extracellular compartment causes a fast increase of postdialysis urea concentration in plasma, i.e., urea rebound (Daugirdas et al., 2001; Daugirdas et al., 2004), Fig. 3. Even if solute removal from a compartment directly accessible to dialyzer is relatively efficient during an intermittent therapy, the overall solute removal may be limited by slow intercompartmental mass transfer. Urea concentration measured in plasma represents the extracellular urea concentration.

The effects of urea generation and urea removal due to solute convective transport that are not included in the basic relation between spKT/V and URR can be corrected by Daugirdas formula (Daugirdas, 1993):

$$\text{spKT/V} = -\ln(\text{R} - 0.008 \cdot \text{T}) + (4 - 3.5 \cdot \text{R}) \cdot \text{UF/W} \tag{19}$$

where T is treatment time in hour, UF is ultrafiltration volume and W is the postdialysis weight (in kilograms). Single-pool kinetics overestimates however the removed amount of urea because of the postdialysis urea rebound, which is an compartmental effect, and therefore the equilibrated KT/V (eqKT/V) was introduced to clinical practice to be estimated by the following formula (Daugirdas et al., 2001):

$$\text{eqKT/V} = -\ln(\mathbb{R}\_{eq} - 0.008 \cdot \text{T}) + (4 - 3.5 \cdot \mathbb{R}\_{eq}) \cdot \text{UF/W} \tag{20}$$

Fig. 3. The phenomena of the intradialytic drop in urea concentration in plasma (inbound), and the postdialysis increase in urea concentration in plasma (rebound).

where

10 Progress in Hemodialysis – From Emergent Biotechnology to Clinical Practice

discrepancy between one and two compartment modeling is larger because the immediate

Assuming one compartment model, a fixed distribution volume (no ultrafiltration) and no generation during the dialysis, as during a short HD session, the concentration of any solute

K t/V CC e t pre

where Ct is the blood concentration of the solute at any time t during dialysis, Cpre is the blood concentration at the beginning of HD, K is the clearance of applied dialyzer, and V is

The single pool KT/V (spKT/V) for urea is determined from equation (15) as the natural logarithm (ln) of the ratio of postdialysis (Cpost) to predialysis (Cpre) plasma urea

spKT/V ln <sup>C</sup>

URR 1 R = − (17)

C R

A solute like urea or creatinine is however removed during hemodialysis more efficiently from the extracellular than from the intracellular compartment and its concentration in plasma falls faster than expected when assessed by one compartment modeling; this effect is called urea inbound (Daugirdas et al., 2001), Fig. 3. When dialysis is completed, the flow from intracellular to extracellular compartment causes a fast increase of postdialysis urea concentration in plasma, i.e., urea rebound (Daugirdas et al., 2001; Daugirdas et al., 2004), Fig. 3. Even if solute removal from a compartment directly accessible to dialyzer is relatively efficient during an intermittent therapy, the overall solute removal may be limited by slow intercompartmental mass transfer. Urea concentration measured in plasma represents the

The effects of urea generation and urea removal due to solute convective transport that are not included in the basic relation between spKT/V and URR can be corrected by Daugirdas

where T is treatment time in hour, UF is ultrafiltration volume and W is the postdialysis weight (in kilograms). Single-pool kinetics overestimates however the removed amount of urea because of the postdialysis urea rebound, which is an compartmental effect, and therefore the equilibrated KT/V (eqKT/V) was introduced to clinical practice to be

spKT /V ln(R 0.008 T) (4 3.5 R) UF/W = − − ⋅+− ⋅⋅ (19)

e q e q eqKT /V ln(R 0.008 T) (4 3.5 R ) UF/W = − − ⋅+− ⋅ ⋅ (20)

post pre

post pre

C

⎛ ⎞ = − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

− ⋅ = ⋅ (15)

<sup>C</sup> <sup>=</sup> (18)

(16)

can be described by the equation (Sargent & Gotch, 1989; Daugirdas et al., 2001):

intercompartmental flow is precluded (Debowska et al., 2007b).

concentrations (Gotch & Sargent, 1985; Daugirdas et al., 2001):

The expression 1 – Cpost/Cpre, is called urea reduction ratio (URR):

estimated by the following formula (Daugirdas et al., 2001):

the solute distribution volume.

extracellular urea concentration.

formula (Daugirdas, 1993):

where

$$\mathbf{R}\_{\text{eq}} = \frac{\mathbf{C}(\mathbf{T}\_{\text{eq}})}{\mathbf{C}\_0} \tag{21}$$

C(Teq) is the urea concentration 30 to 60 minutes after the dialysis session. The eqKT/V is typically about 0.2 KT/V unit lower than the spKt/V, but this difference depends on the efficiency, or rate of dialysis (Daugirdas et al., 2001). Equilibrated KT/V values can be also calculated using an alternative equation, as described by Daugirdas and Schneditz (Daugirdas & Schneditz, 1995):

$$\text{eqKT/V} = \text{spKT/V} - 0.6 \cdot \frac{\text{spKT/V}}{\text{T}} + 0.03 \tag{22}$$

or the formula derived from observations during the HEMO Study (Depner et al., 1999; Eknoyan et al., 2002; Daugirdas et al., 2004):

$$\text{eqKT/V} = \text{spKT/V} - 0.39 \cdot \frac{\text{spKT/V}}{\text{T}} \tag{23}$$

or that introduced by Tattersall et al. (Tattersall et al., 1996):

$$\text{eqKT} / \text{V} = \text{spKT} / \text{V} \cdot \frac{\text{T}}{\text{T} + \text{36}} \tag{24}$$

where T indicates treatment time in minutes. Equations (22) and (23), were derived from regression using the rebounded BUN measured 30 or 60 minutes after dialysis. The Tattersall equation was derived from theoretical considerations of disequilibrium and rebound, but the coefficient was derived from fitting to clinical data.

Kinetic Modeling and Adequacy of Dialysis 13

<sup>G</sup> EKR

The equation (30) may be used in metabolically stable patients, whereas in acute renal failure patients the definition for EKR requires a more unifying form (Casino & Marshall, 2004):

<sup>Δ</sup>M /T EKR

where ΔMR is total solute amount removed by replacement therapy and the kidneys, and T is arbitrary assumed time. EKR, in the form of equation (31), is determined as solute

Taking into account the average predialysis urea concentration, Gotch introduced the standard KT/V (stdKT/V) concept to measure the relative efficiency of the whole spectrum of dialytic therapies whether intermittent, continuous or mixed (Gotch, 1998). The stdKT/V was defined with a relation between urea generation, expressed by its equivalent normalized protein catabolic rate (nPCR) and the peak average urea concentration (Cpa) of

> pa 0.184(nPCR 0.17) V 0.001 7 1440 stdKT /V C V

where 0.184(nPCR – 0.17) V·0.001 is equal to urea generation rate G (mg/min), V is body water in mL and 7·1440 is number of minutes in one week´s time. Predialysis urea concentration (Cpa) - for any combination of frequency of intermittent HD (IHD), automated peritoneal dialysis (APD) and continuous dialysis between IHD or APD sessions - was

( )

1e e

−

spKT /V V /T K K

⋅ + <sup>=</sup>

pa K K 7 / N 1440 T

the equilibrated KT/V calculated according to equation (22).

between stdKT/V, spKT/V and eqKT/V (Leypoldt et al., 2004):

<sup>−</sup> <sup>−</sup>

G G 1e e 1 e

removal rate over time average solute concentration.

all the weekly values (Gotch, 1998; Diaz-Buxo & Loredo, 2006):

**3.4 Standardized KT/V** 

defined as follows (Gotch, 1998):

C

( ) ( )

ta

R ta

<sup>C</sup> <sup>=</sup> (30)

<sup>C</sup> <sup>=</sup> (31)

− ⋅⋅ ⋅ <sup>=</sup> <sup>⋅</sup> (32)

( )( ) ( ) ( )( ) ( )

K K 7 / N 1440 T K K 7 / N 1440 T

p r

p r p r

+ − + −

⎛ ⎞ ⎜ ⎟ ⎝ ⎠

(34)

(33)

( )( ) ( )

+ −

eqKT /V

−

−

eqKT / V V V

− − <sup>−</sup>

p r

− + −

eqKT / V V

where K, Kp and Kr are dialyzer, peritoneal and renal urea clearances, respectively, T is duration of treatment sessions, N is the frequency of IHD or APD per week and eqKT/V is

Assuming a symmetric weekly schedule of dialysis sessions, no residual renal function, and a fixed solute distribution volume V, Leypoldt et al. obtained an analytical relationship

> eqKT /V 1 e <sup>10080</sup>

spKT /V N T

<sup>=</sup> <sup>−</sup> <sup>+</sup> <sup>−</sup> <sup>⋅</sup>

<sup>T</sup> stdKT /V 1 e 10080 <sup>1</sup>

−

### **3.2 Urea KT/V and creatinine clearance for the kidneys**

To assess the residual renal function (RRF) urine is usually collected for 24 hours and analyzed for urea as well as creatinine (Daugirdas et al., 2001). Residual renal clearance for a particular substance can be calculated as follows:

$$\mathbf{K}\_r = \frac{\text{excrection rate}}{\mathbf{C}\_o} = \frac{\mathbf{C}\_{\text{urine}} \mathbf{V}\_{\text{urine}}}{\mathbf{T}\_{\text{urine}}} \frac{\mathbf{1}}{\mathbf{C}\_o} = \frac{\Delta \mathbf{M}\_r}{\mathbf{T}\_{\text{urine}}} \frac{\mathbf{1}}{\mathbf{C}\_o} \tag{25}$$

where Vurine is urine volume, Curine is solute concentration in urine, Turine is time of urine collection, Ce is plasma solute concentration and ΔMr is solute mass removed by the kidneys. Weekly KT/V for the kidney for 1 week time is expressed as follows:

$$\text{weekly (KT/V)}\_{\text{RRF}} = \frac{7 \cdot \mathbb{C}\_{\text{unine}} \text{V}\_{\text{unine}}}{\text{C}\_{\text{o}} \text{V}\_{\text{b}}} = \frac{7 \cdot \Delta \text{M}\_{\text{r}}}{\text{M}\_{\text{b}}} \tag{26}$$

where Mb is solute mass in the body, Vb is TBW and other symbols have the same meaning as in equation (25).

In clinical practice, the most popular methods used for evaluation RRF is creatinine clearance (ClCr), calculated as follows:

$$\text{weekly } \mathbf{Cl}\_{\text{Cr}, \text{RSA}^x} = \frac{7 \cdot \Delta \mathbf{M}\_{\text{R,Cr}}}{1 \,\text{week} \cdot \mathbf{C}\_{\text{o,Cr}}} \frac{1.73}{\text{BSA}} \tag{27}$$

where ΔMR,Cr is creatinine total mass removed during one day due to therapy and by residual renal function, Ce,Cr is serum creatinine concentration, BSA is body surface area and 1.73 is the average BSA for a typical human. Weekly creatinine clearance is the most often expressed in L for 1 week.

#### **3.3 Equivalent renal clearance (EKR)**

In a steady state, during continuous dialytic treatment or/and with renal function, the solute generation rate G is balanced by the solute removal rate Kss determining in this way the constant concentration Css within the patient body (Gotch, 2001):

$$\mathbf{C}\_{\kappa} = \mathbf{G} / \mathbf{K}\_{\kappa} \tag{28}$$

The Kss is defined by rearrangement of equation (28):

$$\mathbf{K}\_{\rm ss} = \mathbf{G} / \mathbf{C}\_{\rm ss} \tag{29}$$

Calculation of a continuous clearance Kss, equivalent to the amount of dialysis provided by any intermittent dialysis schedule, Keq, requires calculation of G and the concentration profile, and selection of a point on this profile, which may be considered to be equivalent to, e.g. weekly, the oscillating concentration (Ceq) according to: Keq = G/Ceq. This approach to the clearance calculation has been reported using different definitions of Ceq. The peak concentration hypothesis defined Ceq as the maximum solute concentration, within e.g. one week duration. The mean predialysis (peak average) solute concentration was used to define standard K (stdK) (Gotch, 1998). The time-average solute concentration (Cta) has been introduced to define 'equivalent renal clearance' (EKR) (Casino & Lopez, 1996):

To assess the residual renal function (RRF) urine is usually collected for 24 hours and analyzed for urea as well as creatinine (Daugirdas et al., 2001). Residual renal clearance for a

excretion rate CV 1 <sup>Δ</sup>M 1 <sup>K</sup>

where Vurine is urine volume, Curine is solute concentration in urine, Turine is time of urine collection, Ce is plasma solute concentration and ΔMr is solute mass removed by the

7C V 7 <sup>Δ</sup><sup>M</sup> weekly KT /V CV M

where Mb is solute mass in the body, Vb is TBW and other symbols have the same meaning

In clinical practice, the most popular methods used for evaluation RRF is creatinine

<sup>7</sup> <sup>Δ</sup><sup>M</sup> 1.73 weekly Cl 1week C BSA

where ΔMR,Cr is creatinine total mass removed during one day due to therapy and by residual renal function, Ce,Cr is serum creatinine concentration, BSA is body surface area and 1.73 is the average BSA for a typical human. Weekly creatinine clearance is the most often

In a steady state, during continuous dialytic treatment or/and with renal function, the solute generation rate G is balanced by the solute removal rate Kss determining in this way

Calculation of a continuous clearance Kss, equivalent to the amount of dialysis provided by any intermittent dialysis schedule, Keq, requires calculation of G and the concentration profile, and selection of a point on this profile, which may be considered to be equivalent to, e.g. weekly, the oscillating concentration (Ceq) according to: Keq = G/Ceq. This approach to the clearance calculation has been reported using different definitions of Ceq. The peak concentration hypothesis defined Ceq as the maximum solute concentration, within e.g. one week duration. The mean predialysis (peak average) solute concentration was used to define standard K (stdK) (Gotch, 1998). The time-average solute concentration (Cta) has been

introduced to define 'equivalent renal clearance' (EKR) (Casino & Lopez, 1996):

kidneys. Weekly KT/V for the kidney for 1 week time is expressed as follows:

RRF

Cr, RRF

the constant concentration Css within the patient body (Gotch, 2001):

The Kss is defined by rearrangement of equation (28):

urine urine r

C T CTC <sup>=</sup> <sup>=</sup> <sup>=</sup> (25)

e b b

R ,Cr

e,Cr

⋅ ⋅ <sup>=</sup> <sup>=</sup> (26)

<sup>⋅</sup> <sup>=</sup> <sup>⋅</sup> (27)

C G/K ss = ss (28)

K G/C ss = ss (29)

e urine e urine e

( ) urine urine <sup>r</sup>

**3.2 Urea KT/V and creatinine clearance for the kidneys** 

particular substance can be calculated as follows:

r

as in equation (25).

expressed in L for 1 week.

clearance (ClCr), calculated as follows:

**3.3 Equivalent renal clearance (EKR)** 

$$\text{EKR} = \frac{\text{G}}{\text{C}\_{\text{ta}}} \tag{30}$$

The equation (30) may be used in metabolically stable patients, whereas in acute renal failure patients the definition for EKR requires a more unifying form (Casino & Marshall, 2004):

$$\text{EKR} = \frac{\Delta \mathbf{M}\_{\text{R}} / \text{T}}{\mathbf{C}\_{\text{ta}}} \tag{31}$$

where ΔMR is total solute amount removed by replacement therapy and the kidneys, and T is arbitrary assumed time. EKR, in the form of equation (31), is determined as solute removal rate over time average solute concentration.

### **3.4 Standardized KT/V**

Taking into account the average predialysis urea concentration, Gotch introduced the standard KT/V (stdKT/V) concept to measure the relative efficiency of the whole spectrum of dialytic therapies whether intermittent, continuous or mixed (Gotch, 1998). The stdKT/V was defined with a relation between urea generation, expressed by its equivalent normalized protein catabolic rate (nPCR) and the peak average urea concentration (Cpa) of all the weekly values (Gotch, 1998; Diaz-Buxo & Loredo, 2006):

$$\text{stdKT} / \text{V} = \frac{0.184 \text{(nPCR} - 0.17) \cdot \text{V} \cdot 0.001}{\text{C}\_{\text{pu}}} \cdot \frac{7 \cdot 1440}{\text{V}} \tag{32}$$

where 0.184(nPCR – 0.17) V·0.001 is equal to urea generation rate G (mg/min), V is body water in mL and 7·1440 is number of minutes in one week´s time. Predialysis urea concentration (Cpa) - for any combination of frequency of intermittent HD (IHD), automated peritoneal dialysis (APD) and continuous dialysis between IHD or APD sessions - was defined as follows (Gotch, 1998):

$$\mathbf{C}\_{\text{pu}} = \frac{\frac{\mathbf{G}}{\left(\text{spKT}/\text{V}\right)\cdot\text{V}/\text{T}} \left(1 - e^{-\text{sp}\mathbf{1}/\text{V}}\right) \mathrm{e}^{\frac{\left(\frac{\mathbf{k}\_{\text{p}} + \mathbf{K}\_{\text{r}}\right)\left(\left(\mathbf{T}/\text{N}\right)\left(\mathbf{M}\cdot\mathbf{0}\right) - \mathbf{I}\right)}{\mathbf{V}}}{\left(1 - e^{-\text{sp}\mathbf{1}/\text{V}}\right) \mathrm{e}^{\frac{\left(\frac{\mathbf{k}\_{\text{p}} + \mathbf{K}\_{\text{r}}\right)\left(\left(\mathbf{T}/\text{N}\right)\left(\mathbf{M}\cdot\mathbf{0}\right) - \mathbf{I}\right)}} + 1} + \frac{\mathbf{G}}{\left(1 - e^{-\frac{\left(\mathbf{k}\_{\text{p}} + \mathbf{K}\_{\text{r}}\right)\left(\left(\mathbf{T}/\text{N}\right)\left(\mathbf{M}\cdot\mathbf{0}\right) - \mathbf{I}\right)}}}{\left(1 - e^{-\text{sp}\mathbf{1}/\text{V}}\right) \mathrm{e}^{\frac{\left(\frac{\mathbf{k}\_{\text{p}} + \mathbf{K}\_{\text{r}}\right)\left(\left(\mathbf{T}/\text{N}\right)\left(\mathbf{M}\cdot\mathbf{0}\right) - \mathbf{I}\right)}} + 1} \right)} \tag{33}$$

where K, Kp and Kr are dialyzer, peritoneal and renal urea clearances, respectively, T is duration of treatment sessions, N is the frequency of IHD or APD per week and eqKT/V is the equilibrated KT/V calculated according to equation (22).

Assuming a symmetric weekly schedule of dialysis sessions, no residual renal function, and a fixed solute distribution volume V, Leypoldt et al. obtained an analytical relationship between stdKT/V, spKT/V and eqKT/V (Leypoldt et al., 2004):

$$\text{stdKT}/\text{V} = \frac{10080 \frac{1 - \text{e}^{-\text{aqK} \cdot \text{V}}}{\text{T}}}{\frac{1 - \text{e}^{-\text{aqK} \cdot \text{V}}}{\text{spKT}/\text{V}} + \frac{10080}{\text{N} \cdot \text{T}} - 1} \tag{34}$$

Kinetic Modeling and Adequacy of Dialysis 15

For the assessment of dialysis efficacy, a few different adequacy indices can be used: a) KT/V (K – dialyzer clearance, T – treatment time, V – solute distribution volume),

There are at least four different reference methods: 1) peak, *p*, 2) peak average, *pa*, 3) time average, *ta*, and 4) treatment time average, *trta*, reference values of concentration, mass and volume, applied in ECC, FSR and KT/V definitions, respectively (ref = p, ref = pa, ref = ta and ref = trta), (Waniewski et al., 2006). For certain applications also minimal average or minimal reference methods are used, e.g. in equation (19) post-dialysis minimal weight is included in calculation of spKT/V. The peak value is the maximal value of solute concentration or mass, the peak average value is calculated as the average of pretreatment values (before each HD session), the time average value is the average calculated over the whole cycle of dialysis, Tc, and the treatment time average value is calculated as the average

Fig. 4. Examples showing urea concentration in extracellular compartment (left side) and urea mass in patient body (right side) during a cycle of three hemodialysis sessions.

**Cta**

**Ctrta**

**Cp Cpa**

The reference solute distribution volume is calculated as the reference mass over the

**10**

**20**

**30**

**Mb, g**

**40**

**41.7**

**50**

Note, that Vref defined in this way may be different from the volume calculated in analogy to Cref or Mb,ref; for example, Vta is in general different from the average volume over the

For HD, dialyzer clearance K is equal to the average effective dialyzer clearance KT defined as solute mass removed from the body during dialysis MRd, per the treatment time, T, and per the average solute concentration in extracellular compartment during treatment time,

Ctrta (K = KT = ΔMRd/T/Ctrta), (Waniewski & Lindholm, 2004; Waniewski et al., 2006). Another concept of clearance, equivalent renal clearance, EKR (mL/min), was proposed by Casino & Lopez for metabolically stable patients, equation (30), but for metabolically unstable patients equation (31) should be used (Casino & Lopez, 1996; Casino & Marshall, 2004), c.f. section 3.3. Using a different concentration in EKR instead of Cta, a general definition of equivalent continuous clearance, ECC, may be formulated (Waniewski et al.,

.

**Time, day**

**Mta**

**Mtrta**

**Mpa Mp**

**0 1 2 3 4 5 6 7** 

**35.1 32.6**

V M /C ref b,ref ref = (35)

**4.1 Different definition variants of KT/V, equivalent continuous clearance (ECC) and** 

b) equivalent continuous clearance, ECC and c) fractional solute removal, FSR.

**fractional solute removal (FSR)** 

reference concentration:

2006; Waniewski et al., 2010), Table 1:

treatment time.

**0.2 0.4 0.6 0.8 1 1.2** 

**Ce, mg/mL** 

**1.12** 

for the time T when dialysis was performed, Fig. 4.

**0.96 0.89**

**0 1 2 3 4 5 6 7**

**Time, day**

where N is number of treatments per week and eqKT/V is derived from spKT/V by using one of the equations (20), (22), (23) or (24). stdKT/V calculated using equation (34) differs slightly from stdKT/V using the exact method, equation (32), that takes into account among other things asymmetry of weekly schedule and Kr (Leypoldt et al., 2004). The stdKT/V is a method to measure the efficiency of HD of variable frequency, continuous peritoneal dialysis (PD), intermittent PD, continuous renal replacement therapies and residual renal function (Diaz-Buxo & Loredo, 2006).
