**4. Integrated system of dialysis adequacy indices**

The integrated system of dialysis adequacy indices aims to include currently applied indices, systemize their definitions and explain relationships between them. The unified approach to the dialysis adequacy 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).

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

An alternative for KT/V is fractional solute removal (FSR), which was suggested by Verrina et al. (Verrina et al., 1998) and Henderson (Henderson, 1999) for comparative studies of different dialysis modalities and schedules. The concept of FSR is closely related to the

SRI was defined for HD as the ratio of net solute removed during a dialysis session (i.e., the solute amount removed minus the solute amount generated in the same time period) over the initial solute amount in the body. This parameter is however useless for comparative analysis of different dialysis modalities and schedules. Its numerical value for the kidneys and continuous therapies, such as continuous ambulatory peritoneal dialysis (CAPD), is by definition equal to zero (Waniewski & Lindholm, 2004). Therefore, Keshaviah (Keshaviah, 1995) used for CAPD and automated peritoneal dialysis the definition of SRI as the ratio of solute removed during a dialysis session over its initial amount in the body, i.e., the

According to the Kidney Disease Outcomes Quality Initiative (KDOQI) guidelines the minimally adequate dose of thrice-weekly HD in patients with residual renal clearance (Kr) less than 2 mL/min/1.73 m2 should be urea single pool KT/V (excluding residual renal function) of 1.2 per dialysis (i.e., an average urea reduction ratio of 65%), (Work Group, 2001). KDOQI Work Group emphasizes that the literature clearly supports the delivery of a minimum hemodialysis dose of at least urea spKt/V = 1.2, but does not suggest an optimal dose. Identification of an optimal dose of hemodialysis would require evaluation of patient status and clinical outcomes including survival analyses and assessment of quality of life as well as the cost-effectiveness of different hemodialysis regimens. Until such data are available, the Work Group states that the hemodialysis dose recommended is to be regarded

The European Best Practice Guidelines recommend higher values: the minimum prescribed HD dose per session for thrice-weekly schedule as equilibrated KT/V for urea is set at 1.2;

The integrated system of dialysis adequacy indices aims to include currently applied indices, systemize their definitions and explain relationships between them. The unified approach to the dialysis adequacy 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;

concept of the solute removal index (SRI) proposed by Keshaviah (Keshaviah, 1995).

**3.5 Solute removal index (SRI) and fractional solute removal (FSR)** 

as a minimum value only (Work Group, 2001; Work Group, 2006).

**4. Integrated system of dialysis adequacy indices** 

this corresponds to a value of spKT/V equal to 1.4 (Work Group, 2002).

function (Diaz-Buxo & Loredo, 2006).

**3.6 International guidelines on HD dose** 

definition of FSR.

Waniewski et al., 2010).

### **4.1 Different definition variants of KT/V, equivalent continuous clearance (ECC) and fractional solute removal (FSR)**

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

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

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.

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

$$\mathbf{V}\_{\text{nof}} = \mathbf{M}\_{\text{b}, \text{nof}} / \mathbb{C}\_{\text{nof}} \tag{35}$$

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 treatment time.

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

$$\text{ECC}\,\text{CC}\_{\text{ref}} = \frac{\Delta\mathbf{M}\_{\text{R}}}{\mathbf{t} \cdot \mathbf{C}\_{\text{ref}}} \tag{36}$$

Kinetic Modeling and Adequacy of Dialysis 17

et al., 2000; Leypoldt et al., 2003; Leypoldt et al., 2004; Waniewski et al., 2006). Both these clearances were defined initially for the metabolic steady state using formula (30), (Casino & Lopez, 1996; Gotch, 1998; Gotch et al., 2000), and were later generalized to the general case

ECC and FSR are not independent indices but they are correlated (Debowska et al., 2005;

ref ref ref <sup>V</sup> ECC FSR

trta ta r c

ref ref ref ref

ΔMR Rd r = ΔM + ΔM , ΔM KTC Rd = trta ⋅ ⋅ , ΔM KTC r r c ta = ⋅ ⋅ (41)

where ECCref and FSRref may be calculated for the same time interval t; a practically important case is t = Tc. The coefficient of proportionality, Vref/t, depends on the choice of reference method, because Vref is defined as Vref = Mb,ref/Cref, equation (35). Furthermore, if t = Tc and the residual renal clearance is Kr, then FSR is related to KT/V (Waniewski et al.,

> C KT C K T FSR CV C V

where ΔMRd and ΔMr are the removed solute mass by replacement therapy and the kidneys, respectively. Another correlation can be found between ECC and K for t = Tc (Waniewski et

> CT C ECC K K CT C

The relationships between ECC and FSR, FSR and KT/V and between ECC and K, equations (39), (40) and (42), respectively, follow directly from their definitions and are valid for all reference methods and any patient and treatment modality (Waniewski et al., 2006). They do not depend on the assumption of the metabolic steady state. However, the coefficients in these relationships, which involve the ratios of different reference concentrations, must be

Different dialysis modalities and schedules are applied in clinics to treat patients with endstage renal diseases. Although solute removal indices are normalized by the solute amount in the body (with the body size included), many other parameters and conditions may differ as the patients are treated by different forms of dialysis (continuous or automated PD, HD, or combination of PD and HD), different number of sessions per week, different duration of each session, and therefore the values of dialysis adequacy indices depend on the details of dialysis. Numerical simulations of different HD regimes were performed using solute kinetic modeling and the obtained solute mass, concentration and distribution volume profiles in body compartments and solute concentration, mass and volume of dialysate were used to calculate dialysis adequacy indices. The two compartment variable volume model,

trta ta ref r ref c ref

<sup>t</sup> <sup>=</sup> (39)

= + (40)

= + (42)

using formula (36), (Casino & Marshall, 2004; Debowska et al., 2010).

ref

calculated for each patient and treatment schedule separately.

**4.2 Typical modalities and schedules for hemodialysis** 

Waniewski et al., 2006):

2006):

because

al., 2006):

where index "ref" denotes a reference concentration, e.g. ref = ta or ref = p, etc. If the patient is in a steady metabolic state, i.e. after a cycle time (Tc) the solute concentration and solute mass in the body return to their initial values, then the total amount of solute removed during Tc is equal to the solute amount generated during Tc. Thus, for the metabolic steady state and t = Tc:

$$
\Delta \mathbf{M}\_{\mathbb{R}} = \mathbf{G} \cdot \mathbf{T}\_{\mathbb{c}} \tag{37}
$$

If one scales the total removed solute mass to some reference mass (Mb,ref) then a nondimensional parameter – fractional solute removal, FSR – may be defined as follows (Gotch, 1998; Waniewski & Lindholm, 2004; Waniewski et al., 2006), Table 1:

$$\text{FSR}\_{\text{ref}} = \frac{\Delta \mathbf{M}\_{\text{R}}}{\mathbf{M}\_{\text{b,ref}}} \tag{38}$$

FSR is often called the solute removal index (SRI), although originally SRI was defined as the solute amount removed minus the solute amount generated in the same time over the initial solute amount in the body, Table 1, (Keshaviah, 1995; Waniewski & Lindholm, 2004, Waniewski et al., 2010).


Table 1. Summary of dialysis adequacy indices.

In particular, EKR is equal to a particular version of ECC (ECCta), equation (36), that was used in many clinical and theoretical studies, Table 1 (Casino & Lopez, 1996; Verrina et al., 1998; Clark et al., 1999; Leypoldt et al., 2003; Casino & Marshall, 2004; Waniewski et al., 2006). If ref = pa (where pa denotes the average predialysis concentration) then ECCpa is equal to stdK defined by Gotch and used in some clinical and theoretical studies, Table 1 (Gotch, 1998; Gotch et al., 2000; Leypoldt et al., 2003; Leypoldt et al., 2004; Waniewski et al., 2006). Both these clearances were defined initially for the metabolic steady state using formula (30), (Casino & Lopez, 1996; Gotch, 1998; Gotch et al., 2000), and were later generalized to the general case using formula (36), (Casino & Marshall, 2004; Debowska et al., 2010).

ECC and FSR are not independent indices but they are correlated (Debowska et al., 2005; Waniewski et al., 2006):

$$\text{ECC}\text{CC}\_{\text{ref}} = \frac{\text{V}\_{\text{ref}}}{\text{t}} \text{FSR}\_{\text{ref}} \tag{39}$$

where ECCref and FSRref may be calculated for the same time interval t; a practically important case is t = Tc. The coefficient of proportionality, Vref/t, depends on the choice of reference method, because Vref is defined as Vref = Mb,ref/Cref, equation (35). Furthermore, if t = Tc and the residual renal clearance is Kr, then FSR is related to KT/V (Waniewski et al., 2006):

$$\text{FSR}\_{\text{ref}} = \frac{\text{C}\_{\text{tra}}}{\text{C}\_{\text{ref}}} \frac{\text{KT}}{\text{V}\_{\text{ref}}} + \frac{\text{C}\_{\text{ta}}}{\text{C}\_{\text{ref}}} \frac{\text{K}\_{\text{r}}\text{T}\_{\text{c}}}{\text{V}\_{\text{ref}}} \tag{40}$$

because

16 Progress in Hemodialysis – From Emergent Biotechnology to Clinical Practice

ref

state and t = Tc:

Waniewski et al., 2010).

*ECC* 

p

pa

ta

<sup>Δ</sup>M /T ECC <sup>C</sup> <sup>=</sup>

<sup>Δ</sup>M /T ECC <sup>C</sup> <sup>=</sup>

stdK, (Gotch, 1998)

<sup>Δ</sup>M /T ECC <sup>C</sup> <sup>=</sup>

Marshall, 2004)

<sup>Δ</sup>M /T ECC <sup>C</sup> <sup>=</sup>

Waniewski et al., 2006)

trta

Table 1. Summary of dialysis adequacy indices.

*Reference method* 

peak, p

peak average, pa

time average, ta

treatment time average, trta

<sup>Δ</sup><sup>M</sup> ECC

where index "ref" denotes a reference concentration, e.g. ref = ta or ref = p, etc. If the patient is in a steady metabolic state, i.e. after a cycle time (Tc) the solute concentration and solute mass in the body return to their initial values, then the total amount of solute removed during Tc is equal to the solute amount generated during Tc. Thus, for the metabolic steady

If one scales the total removed solute mass to some reference mass (Mb,ref) then a nondimensional parameter – fractional solute removal, FSR – may be defined as follows

ref

<sup>Δ</sup><sup>M</sup> FSR =

FSR is often called the solute removal index (SRI), although originally SRI was defined as the solute amount removed minus the solute amount generated in the same time over the initial solute amount in the body, Table 1, (Keshaviah, 1995; Waniewski & Lindholm, 2004,

R

*FSR* 

p

pa

ta

trta

<sup>Δ</sup><sup>M</sup> FSR M=

<sup>Δ</sup><sup>M</sup> FSR M=

<sup>Δ</sup><sup>M</sup> FSR M=

<sup>Δ</sup><sup>M</sup> FSR M<sup>=</sup>

b,ref

(Gotch, 1998; Waniewski & Lindholm, 2004; Waniewski et al., 2006), Table 1:

*Equivalent Continuous Clearance* 

R c

R c

pa

R c

R c

trta

EKR, (Casino & Lopez, 1996; Casino &

K·T/Tc (Lowrie et al., 1999;

In particular, EKR is equal to a particular version of ECC (ECCta), equation (36), that was used in many clinical and theoretical studies, Table 1 (Casino & Lopez, 1996; Verrina et al., 1998; Clark et al., 1999; Leypoldt et al., 2003; Casino & Marshall, 2004; Waniewski et al., 2006). If ref = pa (where pa denotes the average predialysis concentration) then ECCpa is equal to stdK defined by Gotch and used in some clinical and theoretical studies, Table 1 (Gotch, 1998; Gotch

ta

p

R

t C <sup>=</sup> <sup>⋅</sup> (36)

ΔM GT R c = ⋅ (37)

M (38)

*Fractional Solute Removal* 

R

R

b,pa

R

R

K·T/Vtrta (Waniewski et al., 2006)

b,trta

b,ta

stdKT/V (Gotch, 1998)

(Henderson, 1999), SRI, (Keshaviah, 1995)

b,p

ref

$$
\Delta \mathbf{M}\_{\mathrm{R}} = \Delta \mathbf{M}\_{\mathrm{Rd}} + \Delta \mathbf{M}\_{\mathrm{r}} \prime \,\, \Delta \mathbf{M}\_{\mathrm{Rd}} = \mathbf{K} \cdot \mathbf{T} \cdot \mathbf{C}\_{\mathrm{tta}\,\mathrm{r}} \prime \,\, \Delta \mathbf{M}\_{\mathrm{r}} = \mathbf{K}\_{\mathrm{r}} \cdot \mathbf{T}\_{\mathrm{c}} \cdot \mathbf{C}\_{\mathrm{tu}} \tag{41}$$

where ΔMRd and ΔMr are the removed solute mass by replacement therapy and the kidneys, respectively. Another correlation can be found between ECC and K for t = Tc (Waniewski et al., 2006):

$$\text{ECC}\,\mathbf{C}\_{\text{ref}} = \frac{\mathbf{C}\_{\text{tra}}}{\mathbf{C}\_{\text{ref}}} \frac{\mathbf{T}}{\mathbf{T}\_{\text{c}}} \mathbf{K} + \frac{\mathbf{C}\_{\text{ta}}}{\mathbf{C}\_{\text{ref}}} \mathbf{K}\_{\text{r}} \tag{42}$$

The relationships between ECC and FSR, FSR and KT/V and between ECC and K, equations (39), (40) and (42), respectively, follow directly from their definitions and are valid for all reference methods and any patient and treatment modality (Waniewski et al., 2006). They do not depend on the assumption of the metabolic steady state. However, the coefficients in these relationships, which involve the ratios of different reference concentrations, must be calculated for each patient and treatment schedule separately.

### **4.2 Typical modalities and schedules for hemodialysis**

Different dialysis modalities and schedules are applied in clinics to treat patients with endstage renal diseases. Although solute removal indices are normalized by the solute amount in the body (with the body size included), many other parameters and conditions may differ as the patients are treated by different forms of dialysis (continuous or automated PD, HD, or combination of PD and HD), different number of sessions per week, different duration of each session, and therefore the values of dialysis adequacy indices depend on the details of dialysis. Numerical simulations of different HD regimes were performed using solute kinetic modeling and the obtained solute mass, concentration and distribution volume profiles in body compartments and solute concentration, mass and volume of dialysate were used to calculate dialysis adequacy indices. The two compartment variable volume model,

Kinetic Modeling and Adequacy of Dialysis 19

**HD3x**

**0.65**

**0.50**

**0.36**

**0.22**

**0.49**

**0.62**

**0.83**

**0.70**

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

**Time, day**

**HD6xd**

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

**Time, day**

**HD6xn**

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

**Time, day**

Fig. 5. Urea concentration, Ce, in the extracellular compartment during conventional hemodialysis provided three times a week (HD3x), daily hemodialysis carried out six times a week (HD6xd) and long, nocturnal hemodialysis (HD6xn). Average urea concentration

**0.28 0.27 0.27 0.27**

**0.45 0.43 0.41 0.41**

**0**

**1**

**0**

**1**

**0**

**0.2**

**0.4**

**0.49**

**0.33**

**0.6**

**Ce, mg/mL**

was plotted with dashed line.

**0.8**

**0.2**

**0.4**

**0.6**

**Ce, mg/mL**

**0.62**

**0.51**

**0.8**

**0.2**

**0.4**

**0.6**

**Ce, mg/mL**

**0.8**

**0.83**

**1**

equation (8), was implemented in the computer program Matlab and solved by numerical integration (Runge-Kutta method) to describe the solute and fluid transport between patient and removal device during dialysis.

### **4.3 Comparison of adequacy indices for different HD regimes based on computer simulations**

The objective of the analysis presented here was to compare different adequacy parameters and their different definitions for different schedules of HD, Table 2:


Values of HD duration and dialyzer clearance were taken to be the average for patients groups enrolled in the Frequent Hemodialysis Network Daily and Nocturnal clinical trails (Daugirdas et al., 2010). Computer simulations were carried out for several weeks of the treatment to achieve the metabolic steady state of the patient.


Table 2. Time schedule and dialyzer clearance K for: conventional hemodialysis provided three times a week (HD3x), daily hemodialysis carried out six times a week (HD6xd) and long, nocturnal hemodialysis (HD6xn).

Other parameters were: urea generation rate, G = 7 mg/min, residual urea clearance Kr = 0.6 mL/min. The convective transport of the solute was characterized by transmittance coefficient, Tr = 0.3, equation (5), for hemodialyzer. For the two compartment model, it was assumed that the intercompartmental clearance Kc = 600 mL/min and volumes of extracellular and intracellular compartments were changed according to equation (10) with α = 1/3. The postdialysis water distribution volume was Vb = 40 L; water was generated with constant rate (Gw = 1.04 mL/min); weekly 10.5 L of water was removed by means of residual water clearance (Krw = 0.1 mL/min) and as a result of ultrafiltration Qv.

The changes of urea concentration in the extracellular compartment of the body and the values of FSR, as obtained by computer simulations using parameters from Table 2, were shown in Fig. 5 and Fig. 6. The time average concentration, Cta, was 0.5 mg/mL in conventional HD performed three times per week and 0.36 mg/mL and 0.22 mg/mL for daily and nocturnal HD carried out six times per week, respectively, Fig. 5 and Table 3. The amplitude of urea concentration changes had the highest values for HD3x and the lowest for HD6xn, Fig. 5 and Table 3.

The weekly values of ECC and FSR, according to all methods for the definition of reference values, equations (36) and (38), and the respective values of urea concentrations in blood, Cref, are shown in Table 3. The adequacy indices were different, with the indices ECC and FSR for HD3x being lower than for HD6xd and HD6xn, Table 3.

The adequacy indices, ECC and FSR, had the highest values for the definitions based on treatment time (trta) reference method and the lowest values for the definitions based on the peak reference method (Table 3), and were between weekly ECCta = 14.03 mL/min and

equation (8), was implemented in the computer program Matlab and solved by numerical integration (Runge-Kutta method) to describe the solute and fluid transport between patient

The objective of the analysis presented here was to compare different adequacy parameters

Values of HD duration and dialyzer clearance were taken to be the average for patients groups enrolled in the Frequent Hemodialysis Network Daily and Nocturnal clinical trails (Daugirdas et al., 2010). Computer simulations were carried out for several weeks of the

*Label Time schedule K, mL/min HD3x* 3 x 219 min 272 *HD6xd* 6 x 147 min 277 *HD6xn* 6 x 401 min 170 Table 2. Time schedule and dialyzer clearance K for: conventional hemodialysis provided three times a week (HD3x), daily hemodialysis carried out six times a week (HD6xd) and

Other parameters were: urea generation rate, G = 7 mg/min, residual urea clearance Kr = 0.6 mL/min. The convective transport of the solute was characterized by transmittance coefficient, Tr = 0.3, equation (5), for hemodialyzer. For the two compartment model, it was assumed that the intercompartmental clearance Kc = 600 mL/min and volumes of extracellular and intracellular compartments were changed according to equation (10) with α = 1/3. The postdialysis water distribution volume was Vb = 40 L; water was generated with constant rate (Gw = 1.04 mL/min); weekly 10.5 L of water was removed by means of

The changes of urea concentration in the extracellular compartment of the body and the values of FSR, as obtained by computer simulations using parameters from Table 2, were shown in Fig. 5 and Fig. 6. The time average concentration, Cta, was 0.5 mg/mL in conventional HD performed three times per week and 0.36 mg/mL and 0.22 mg/mL for daily and nocturnal HD carried out six times per week, respectively, Fig. 5 and Table 3. The amplitude of urea concentration changes had the highest values for HD3x and the lowest for

The weekly values of ECC and FSR, according to all methods for the definition of reference values, equations (36) and (38), and the respective values of urea concentrations in blood, Cref, are shown in Table 3. The adequacy indices were different, with the indices ECC and

The adequacy indices, ECC and FSR, had the highest values for the definitions based on treatment time (trta) reference method and the lowest values for the definitions based on the peak reference method (Table 3), and were between weekly ECCta = 14.03 mL/min and

residual water clearance (Krw = 0.1 mL/min) and as a result of ultrafiltration Qv.

FSR for HD3x being lower than for HD6xd and HD6xn, Table 3.

**4.3 Comparison of adequacy indices for different HD regimes based on computer** 

and their different definitions for different schedules of HD, Table 2:

2. Daily hemodialysis with six 147-minute sessions (*HD6xd*) 3. Nocturnal hemodialysis with six 401-minute sessions (*HD6xn*)

treatment to achieve the metabolic steady state of the patient.

1. Conventional, daily hemodialysis with three 219-minute sessions (*HD3x*)

and removal device during dialysis.

long, nocturnal hemodialysis (HD6xn).

HD6xn, Fig. 5 and Table 3.

**simulations** 

Fig. 5. Urea concentration, Ce, in the extracellular compartment during conventional hemodialysis provided three times a week (HD3x), daily hemodialysis carried out six times a week (HD6xd) and long, nocturnal hemodialysis (HD6xn). Average urea concentration was plotted with dashed line.

Kinetic Modeling and Adequacy of Dialysis 21

FSRpa = 2.26 for HD3x and weekly ECCta = 32.3 mL/min and FSRpa = 5.44 for HD6xn, indicating more efficient solute removal with HD6xn. The difference between the values of the indices calculated according to different definitions (treatment time average, time

The ratio of ECC and FSR differed slightly between the modalities and definitions (range, 4.04 - 4.96 mL/min) and correlated with the fluctuations of water volume and urea concentration in the body, as shown by Vref, Table 3. Nevertheless, equation (39) is valid for all investigated applications. Because the cycle time was the same for all simulated dialysis modalities, Tc = 1 week, thus the correlation between the ratio of ECC to FSR and water

 *ECC FSR* ECC

*HD3x* p 8.42 1.94 4.34 43.76 0.83

*HD6xd* p 11.28 (34%) 2.69 (39%) 4.20 42.32 0.62

*HD6xn* p 14.25 (69%) 3.42 (76%) 4.17 41.99 0.49

Table 3. Weekly ECC, FSR, the ratio of ECC to FSR, the solute distribution volume, Vref, and urea concentration in extracellular compartment, Cref, calculated according to four different definitions: peak (p), peak average (pa), time average (ta) and treatment time average (trta) for conventional hemodialysis provided three times a week (HD3x), daily hemodialysis carried out six times a week (HD6xd) and long, nocturnal hemodialysis (HD6xn). Values in

The formula for the relationship between FSR and KT/V, equation (40), shows that FSR may be represented as a weighted sum of KT/V and KrTc/V, with the first term representing the urea removal by dialysis and the second one, the urea removal by residual clearance. The weighing coefficients are the ratios of the average urea concentration in blood during dialysis treatment over the reference urea concentration and the average urea concentration in blood during the whole treatment cycle over the reference concentration, respectively, Table 4. These coefficients depend on the reference method as well as the treatment

ECC may be related to K and Kr using equation (42). For that purpose K must be recalculated by the factor T/Tc, and then the recalculated value of K and the value of Kr are summed up with the same weighing coefficients that appear in formula (40) for the relationship of FSR and KT/V. The weighing coefficients show how much the average concentrations, during effective treatment time T, and during the whole cycle time Tc,

brackets present the difference in relation to conventional HD (in percent).

respectively, differ from the reference concentration, Table 4.

modality and schedule.

pa 9.62 2.26 4.25 42.85 0.73 ta 14.03 3.38 4.16 41.90 0.50 trta 18.83 3.80 4.96 50.01 0.37

pa 14.84 (54%) 3.64 (61%) 4.08 41.11 0.47 ta 19.71 (40%) 4.85 (43%) 4.07 40.99 0.36 trta 25.31 (34%) 5.26 (38%) 4.81 48.46 0.28

pa 21.94 (128%) 5.44 (141%) 4.04 40.69 0.32 ta 32.3 (130%) 7.95 (135%) 4.06 40.93 0.22 trta 41.66 (121%) 9.35 (146%) 4.45 44.90 0.17

FSR

*Vref Cref*

average, peak average, peak) was high (up to 192%).

volume confirmed the relationship described by equation (39).

Fig. 6. FSR, normalized by peak, p, peak average, pa, time average, ta, and treatment time average, trta, urea mass in the body during conventional hemodialysis provided three times a week (HD3x), daily hemodialysis carried out six times a week (HD6xd) and long, nocturnal hemodialysis (HD6xn).

**HD3x**

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

**Time, day**

**HD6xd**

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

**Time, day**

**HD6xn**

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

**Time, day**

Fig. 6. FSR, normalized by peak, p, peak average, pa, time average, ta, and treatment time average, trta, urea mass in the body during conventional hemodialysis provided three times

a week (HD3x), daily hemodialysis carried out six times a week (HD6xd) and long,

**p**

**p**

**trta ta**

**p**

**pa**

**trta ta pa**

**trta ta pa**

**0**

**0**

**0**

nocturnal hemodialysis (HD6xn).

**2**

**4**

**6**

**FSR**

**8**

**10**

**2**

**4**

**6**

**FSR**

**8**

**10**

**2**

**4**

**6**

**FSR**

**8**

**10**

FSRpa = 2.26 for HD3x and weekly ECCta = 32.3 mL/min and FSRpa = 5.44 for HD6xn, indicating more efficient solute removal with HD6xn. The difference between the values of the indices calculated according to different definitions (treatment time average, time average, peak average, peak) was high (up to 192%).

The ratio of ECC and FSR differed slightly between the modalities and definitions (range, 4.04 - 4.96 mL/min) and correlated with the fluctuations of water volume and urea concentration in the body, as shown by Vref, Table 3. Nevertheless, equation (39) is valid for all investigated applications. Because the cycle time was the same for all simulated dialysis modalities, Tc = 1 week, thus the correlation between the ratio of ECC to FSR and water volume confirmed the relationship described by equation (39).


Table 3. Weekly ECC, FSR, the ratio of ECC to FSR, the solute distribution volume, Vref, and urea concentration in extracellular compartment, Cref, calculated according to four different definitions: peak (p), peak average (pa), time average (ta) and treatment time average (trta) for conventional hemodialysis provided three times a week (HD3x), daily hemodialysis carried out six times a week (HD6xd) and long, nocturnal hemodialysis (HD6xn). Values in brackets present the difference in relation to conventional HD (in percent).

The formula for the relationship between FSR and KT/V, equation (40), shows that FSR may be represented as a weighted sum of KT/V and KrTc/V, with the first term representing the urea removal by dialysis and the second one, the urea removal by residual clearance. The weighing coefficients are the ratios of the average urea concentration in blood during dialysis treatment over the reference urea concentration and the average urea concentration in blood during the whole treatment cycle over the reference concentration, respectively, Table 4. These coefficients depend on the reference method as well as the treatment modality and schedule.

ECC may be related to K and Kr using equation (42). For that purpose K must be recalculated by the factor T/Tc, and then the recalculated value of K and the value of Kr are summed up with the same weighing coefficients that appear in formula (40) for the relationship of FSR and KT/V. The weighing coefficients show how much the average concentrations, during effective treatment time T, and during the whole cycle time Tc, respectively, differ from the reference concentration, Table 4.

Kinetic Modeling and Adequacy of Dialysis 23

method should be used (ref = p, ref = pa, ref = ta or ref = trta) for the assessment of the treatment adequacy. To get a consistent scheme of definitions and relationships, the reference solute distribution volume was defined as Vref = Mb,ref / Cref. For each reference method, three adequacy indices, FSR, KT/V and ECC, can be defined. The computer simulations demonstrated that these indices are related, and that the relationships follow

In general, ECC is equivalent to FSR, equation (39), if the same type of reference method is applied for both parameters (Debowska et al., 2005; Waniewski et al., 2006). The coefficient of proportionality, Vref/Tc, depends only slightly on the details of the procedure, especially on the schedule of water removal and the degree of total body water variation during the treatment cycle as well as the difference between urea concentrations in intracellular and extracellular compartments that may develop during dialysis sessions. Nevertheless, the variations of Vref between different definitions and procedures for the same patient are small. If a reference method (p, pa, ta, trta) of FSR and ECC definitions is fixed, then the changes in FSR are reflected by the changes in ECC and vice versa for the same patient. However, this relationship is different for patients with different total body water, which

One advantage of using equivalent continuous clearance, ECC, or fractional solute removal, FSR, is that these indices permit comparison of hemodialysis and peritoneal dialysis doses, and allow the addition of the contributions from HD, PD and residual renal function into the whole index for solute removal efficiency, and thus these indices could provide a basis for setting one standard target dose for all patients regardless of dialysis modality, frequency and duration (Depner, 2005; Debowska et al., 2007a). Note that ECC and FSR may also be successfully applied in continuous and semi-continuous therapies (e.g. continuous veno-venous hemofiltration, CVVH, slow low-efficiency daily dialysis, SLEDD) in patients with acute renal failure (Clark et al., 1999; Leypoldt et al., 2003; Debowska et al., 2010). From the beginning of the era of dialysis treatment, there has been a quest for the optimal dialysis index. The history reflects the complexity of this matter, and attempts to simplify the meander way of this process that has not yet been finished because different versions of existing dialysis modalities are applied, new therapies are being introduced into clinical practices as new techniques become available. Compartmental models and solute kinetic analysis, presented here, used for the mathematical and computer-based description of

delivered dose of dialysis are important tools for the evaluation of dialysis adequacy.

Canaud, B., Garred, L. J., Argiles, A., Flavier, J. L., Bouloux, C. & Mion, C. (1995). Creatinine

Casino, F. G. & Lopez, T. (1996). The equivalent renal urea clearance: a new parameter to assess dialysis dose. *Nephrol Dial Transplant*, Vol. 11, No. 8, pp. (1574-81) Casino, F. G. & Marshall, M. R. (2004). Simple and accurate quantification of dialysis in

kinetic modelling: a simple and reliable tool for the assessment of protein nutritional status in haemodialysis patients. *Nephrol Dial Transplant*, Vol. 10, No. 8,

acute renal failure patients during either urea non-steady state or treatment with irregular or continuous schedules. *Nephrol Dial Transplant*, Vol. 19, No. 6, pp. (1454-

their definitions.

**7. References** 

66)

pp. (1405-10)

may also differ between patient populations.


Table 4. Nondimensional parameters KT/Vref, residual KrTc/Vref, the ratio of treatment time average to reference urea concentration Ctrta/Cref, the ratio of time average to reference urea concentration Cta/Cref and fractional solute removal, FSR, equation (40), for conventional hemodialysis provided three times a week (HD3x), daily hemodialysis carried out six times a week (HD6xd) and long, nocturnal hemodialysis (HD6xn).

ECC and FSR were found to be equivalent descriptions of dialysis, if the same reference method (peak, peak average, time average, treatment time average) was used, as suggested by equation (39). The ratio of ECC and FSR was similar for all definitions, in contrast to much different values of the indices themselves.
