Pediatric Nephrology

Pediatr Nephrol (1992) 6: 349- 353 9 IPNA 1992

Original article Mathematical modelling of haemodialysis in children Jonathan H. C. Evans1, Stephen W. Smye2, and J. Trevor Brocklebank 1 Departments of 1Paediatrics and Child Health and 2Medical Physics, St Jamcs's University Hospital, Leeds LS9 7TF, UK Received December 21, 1990, received in revised form November 29, 1991; accepted December 3, 1991

Abstract. The single-pool urea kinetic model (UKM), utilising "Kt/V" (the normalised whole body urea clearance), is widely used to help assess the adequacy of haemodialysis in adults. In the presence of an adequate dietary protein intake, a value of unity is acceptable for thrice weekly dialysis. Children could benefit from this approach but, with their relatively higher protein intakes and dialysis needs, this model may not be applicable. Urea kinetics, studied in six children with chronic renal failure by serial timed blood urea measurements during and after haemodialysis, were compared with the kinetics of a one-pool and a two-pool UKM. The two-pool UKM with intra- and extracellular pools best fitted the observed data, re-equilibration between pools accounting for the marked rebound increase in blood urea seen in the 1st h after dialysis (kt 17%, SD 5). Kt/V calculated using the end-dialysis blood urea was higher (Ix 21%, SD 5) than when the more correct equilibrated value was used. The post-dialysis rebound indicates significant disequilibrium between the two pools at the end of dialysis. Dialysis efficiency may be substantially overestimated unless this is allowed for by using the rebounded post-dialysis blood urea when calculating Kt/V. Key words: Urea kinetics - Haemodialysis - Two-pool model - Chronic renal failure - Disequilibrium

Introduction When assessing the adequacy of dialysis, the removal of urea is only one of a wide range of factors, such as acidbase balance, fluid removal and blood pressure control, that need consideration. Nonetheless, the single-pool urea kinetic model (UKM) is now widely used to manage adults

Correspondence to: J. H. C. Evans, University Department of Child Health, Royal Manchester Children's Hospital, Pendelbury, Manchester, M27 1HA, UK

on maintenance haemodialysis [ 1], where measurement of the protein catabolic rate (PCR) and of the Kt/V, the normalised whole body urea clearance (see appendix), has allowed individualised prescription of dialysis, matching diet and dialysis needs [2]. It also enables the physician to easily discriminate between dietary and dialysis problems when faced with unsatisfactory biochemistry, and to identify underdialysed patients [3], whilst dieticians can focus their attention on patients with unsatisfactory PCRs in whom diet can be more closely monitored. By prescribing dialysis in this manner morbidity can be reduced [2]. Although children on haemodialysis suffer all the problems of adults, urea kinetic modelling of paediatric dialysis has received far less attention and dialysis prescription is often on rather more empirical grounds [4-6]. Before UKM can properly be applied to children, any model to be used needs validating. Movement of urea between intracellular fluid (ICF) and extracellular fluid (ECF) is limited by the mass transfer coefficient for urea [7]. In modelling adult dialysis on a single pool this process is ignored because the mass transfer coefficient is considerably higher than the dialyser urea clearance [7], and hence urea removal from the ECF by dialysis is easily matched by movement out of the ICF. Children, however, when considered on a weight for weight basis, have higher metabolic rates and nutritional requirements [5], hence dialysis needs are greater and relatively high urea clearances are used that may result in greater disequilibrium between ICF and ECF urea concentrations during haemodialysis. A two-pool UKM, with ICF and ECF compartments, would allow for this disequilibium and enable more accurate study and understanding of dialysis kinetics in children, perhaps resulting in improved dialysis regimens. This study was performed to determine the amount of intercompartmental disequilibrium of urea that occurs during haemodialysis in children.

Patients and methods Six children, aged 4-16 years (median 7 years) and weighing 11-31 kg (median 20 kg), with stable chronic renal failure, from renal dysplasia

350 35

3~1

30,

30,

25

25

-~ 20 E E o 15

20

9

Finish

Finish

E E ~15 .=

dJ

lO

10

0

0

I

I

I

I

50

100

150

200

0

250

I

50

Time (rnin)

I

I

I

100 150 Time (rnin)

200

250

Fig. 1. Blood urea in one child during and after the finish of haemodialysis (-A-) compared with the values predicted by the single-pool urea kinetic model (UKM) in the same child ( - . - ) assuming a total body water of 0.61/kg

Fig. 2. Blood urea in one child during and after the finish of haemodialysis (-A-) compared with the values predicted by the two-pool UKM (--X--) in the same child assuming intracelhilar fluid (ICF) and extracellular fluid (ECF) volumes of 0.4 and 0.21/kg, respectively

(3), reflux nephropathy (2) and glomerulonephritis, were studied during a single dialysis session of 2 - 3 h. A Gambro AK10 machine was used, with hollow-fibre dialysers (Nephross Lento of Nipro FB series), vascular access was through a dual-lumen subclavian catheter (Quinton Permcath) with blood flow rates of 100-150 ml/min; recirculation, measured in three subjects, was less than 2%. Ultrafiltration was carded out prior to dialysis in three subjects; the amount of urea removed in this manner was small (5%- 10%) in relation to dialysis and was therefore not included in subsequent calculations. In each subject, a concentration-time profile for blood urea was constructed during dialysis and for 1 h afterwards, dialyser urea clearance was measured and the mass transfer coefficient calculated. With this information the observed urea kinetics were compared with those predicted by a single-pool UKM and by a two-pool UKM (see appendix). Blood samples were taken from the arterial limb of the dialysis curcuit every 15 min throughout dialysis, and at 5, 10, 20, 30, 40, 50 and 60 rain after dialysis. Blood urea measurements were performed in a single batch using a Parallel Autoanalyser (American Monitor Company, Burgess Hill, W. Sussex, UK). Residual renal function was assessed by urea clearance from a timed interdialytic urine collection with blood samples at the start and finish of the collection. Urea clearance for the dialyser was calculated from paired blood samples from the arterial and venous limb of the dialyser circuit using the standard arterial-venous clearance formula [8]

Table 1. Urea clearance in the six patients calculated using arterialvenous (A-V) clearance formula and dialysate formula

clearance = Qb -

A-V A

where Qb = blood flow and A - V = arterial-venous urea concentration, taking the mean of three such measurements. Blood flow pumps were carefully calibrated prior to the study. A second estimate of urea clearance was obtained simultaneously from 20-ml aliquots of dialysate collected from the dialysate effluent pipe, using the formula D clearance = Qd " - A where Qd = dialysate flow rate (ml/min) timed over i min and D = dialysate urea concentration. Dialysate flow rates were between 510 and 545 ml/min. Urea generation rates (UGRs) were estimated from the increase in blood urea between dialyses, assuming a volume of distribution of 0.6 l/kg body weight, plus the urinary excretion of urea, which was

1

2

3

4

5

6

A - V clearance (ml/min)

61

72

61

89

102

101

Dialysate clearance ml/min)

60

-

58

90

94

109

measured in a timed collection over the same interdialytic time period. The PCR was calculated from the UGR using the formula [9] PCR (g/day) = 9.35 UGR (mmol/h) +0,17 weight (kg)

Results

Residual renal function, assessed by urinary urea clearance was l o w i n five c h i l d r e n (0, 0, 0.6, 0.6, 1.6 m l / m i n ) , m e a s u r e m e n t w a s n o t p o s s i b l e i n o n e child w i t h a c o l o n i c c o n d u i t . T h e r e w a s c l o s e a g r e e m e n t b e t w e e n the v a l u e s for u r e a c l e a r a n c e o b t a i n e d f r o m the A - V m e t h o d a n d the d i a l y s a t e m e t h o d ( T a b l e 1). The change in blood urea seen during dialysis in one subject, r e p r e s e n t a t i v e o f the g r o u p , is c o m p a r e d w i t h the k i n e t i c s p r e d i c t e d b y the s i n g l e - p o o l a n d t w o - p o o l U K M i n Figs. 1 a n d 2, respectively. T h e s i n g l e - p o o l m o d e l prov i d e d a c l o s e r fit i n m o s t cases. H o w e v e r , m a n i p u l a t i o n o f the t w o - p o o l m a t h e m a t i c a l m o d e l b y i n c r e a s i n g the E C F v o l u m e to b e t w e e n 0.25 a n d 0.3 l a g r e s u l t e d i n the t w o p o o l m o d e l p r o v i d i n g a m u c h c l o s e r fit i n e a c h s u b j e c t (Fig. 3). I n the 1st h after dialysis there w a s a s u b s t a n t i a l r e b o u n d i n c r e a s e i n b l o o d u r e a o v e r the v a l u e at the e n d o f d i a l y s i s ( T a b l e 2). T h e m e a n i n c r e a s e w a s 17% ( S D 5%). T h e

351

11

35

30, 10.5

25

-6 E

-~ 20 Finish

E E

9"

15

-

~

E lO

~

t..

10 9.5

0

0

I

I

I

I

50

100

150

200

Time(rnin)

250

0

I

I

I

I

I

I

10

20

30

40

50

60

70

Time(rnin)

Fig. 3. Blood urea in one child during and after the finish of haemodialysis (-A-) compared with the values predicted by the two-pool UKM (---~--) in the same child assuming ICF and ECF volumes of 0.4 and 0.3 l/kg, respectively

Fig. 4. The rebound increase in blood urea in the 1st h after the finish of haemodialysis (--~-) in one subject with the best fit exponential curve (R2 = 0.98)

Table 2. Urea kinetic data calculated from the changes in blood urea, during, immediately after and between dialysis sessions Patient no.

Rebound increase in blood urea (%)

Mass transfer coefficient (ml/min per kg)

Dialyser urea clearance (ml/min per kg)

Rate of urea generation (mmol/kg per hour)

Protein catabolic rate (g/kg per day)

Ultrafiltration rate during dialysis (ml/min)

1

16 18 8 22 17 23

7.4 6.2 8.9 7.6 9.1 8.8

5.7 5.2 3.4 4.2 4.6 3.4

0.15 0.16 0.17 0.14 0.26 0.29

1.6 1.7 1.8 1.5 2.6 2.8

0.0 1.7 1.0 4.6 4.0 4.9

2 3 4 5 6

Table 3. The effect of disequilibrium on the calculation of normalised whole body urea clearance (Kt/V) using the immediate and equilibrated (1 h post dialysis) blood urea value Patient no.

Kt/V equilibrated

Kt/V immediate

% increase

1 2 3 4 5 6

0.96 1.03 0.46 0.59 0.71 0.73

1.10 1.19 0.54 0.73 0.88 0.94

15 16 17 24 24 28

observed values closely fitted an exponential curve (Fig. 4) with R2>0.97 in each case, in keeping with the re-equilibration phase predicted by the two-pool UKM. The mass transfer coefficient for urea calculated from the re-equilibration is shown in Table 2, the mean value was 8.0 ml/min per kg (SD 1.1). The ultrafiltration rate and the estimates for UGR and PCR are also shown in Table 2. Shown in Table 3 are the values for Kt/V obtained from Eq. 2 (see appendix) using pre- and immediately postdialysis urea compared with the values obtained if the equilibrated post-dialysis urea is used. The values are

higher (mean 21%, SD 5) when based on the immediate sample.

Discussion

The purpose of this study was to determine the magnitude of intercompartmental disequilibrium of urea present when children undergo haemodialysis, and to assess its potential influence on the estimation of dialysis efficiency based on the difference in blood urea concentration before and after dialysis. With this in mind, a number of simplifications have been deliberately made in the models in order to emphasise the main features and to reduce the number of unknown parameters. Thus the effect of ultrafiltration was neglected as this process contributes little towards the clearance of urea [1]; for example, patient no. 6 had the highest ultrafiltration rate (5 ml/min) and midification to allow for its contribution [1] increases the urea clearance from 106 ml/min to 107 ml/min, a change of less than 1%. Urea generation was neglected as its magnitude is small when compared with the removal of urea during dialysis. For example, the UGRs shown in Table 1 represent a rise in blood urea of between 7 and 20 mmol/l between consec-

352

utive dialyses, a time period of 45 h, which results in a rate of increase of 0.15-0.45 mmol/l per hour. This is small in comparison to the rebound increase in blood urea at the end of dialysis which was between 1.0 and 2.8 mmol/1 in the 1st h. The validity of the model is supported by the similar values for urea clearance (Table 1) obtained from two independent methods; the removal of urea from the blood ( A - V formula) and the appearance of urea in dialysate. More rigorous methods to ensure accurate clearance values would be preferable if UKM is used clinically [ 10]. Although these simplifications are appropriate in this study, when UKM is used in clinical practice the adequacy of dialysis can only be assessed with knowledge of the dietary protein intake, and in the context of other indices of renal failure. The changes in blood urea seen during dialysis could be fitted to either of the models but the rebound increase in urea after dialysis was consistent only with the two-pool model. In order to improve the agreement between the experimental data and the model predictions it was necessary to increase the ECF volume from the initial estimate of 0.2 1/kg [11]. This change is supported by data showing that the majority of children on haemodialysis had a postdialysis ECF volume in the range 0.22-0.29 l/kg, [12], volumes that are consistent with the values of 0.250.30 l&g that were necessary to fit the two-pool UKM to the data in this study (exemplified in Fig. 3). Two explanations have been offered for the rebound increase in blood urea seen at the end of dialysis: (1) that it is the result of increased protein catabolism during dialysis triggered by patient-dialyser bioincompatibility [13, 14] and (2) that it is the result of re-equilibration between the ICF and ECF [15]. Although both phenomena may coexist, the latter explanation seems most plausible for the early exponential rebound, which is known to be related to the magnitude of the dialyser urea clearance rather than to indices of bioincompatibility [15]. The data in this study indicate an exponential rebound in keeping with re-equilibration. From these data values for the transcellular mass transfer coefficient have been derived which, allowing for the size of our subjects, are in keeping with published values from adults [15, 16]. Many adult dialysis units are now using urea kinetic modelling to monitor and prescribe dialysis using commer, cial software packages (e. g. Theraps, Cobe Laboratories, Lakewood, Colorado, USA) based on single-pool urea kinetics. Adequacy of dialysis is assessed by the value of Kt/V, calculated from pre- and post-dialysis blood urea. In the presence of a satisfactory protein intake, a value of Kt/V between 0.9 and 1.5 is considered acceptable [2]. Although some paediatric centres have adopted this approach [9], there are as yet no guidelines for a target range of Kt/V in children. One potential error in the calculation of Kt/V (Eq. 2) is the presence of disequilibrium at the end of dialysis, with the ECF urea concentration lower than the ICF value. Thus the post-dialysis blood urea concentration will underestimate the concentration of urea within the body pool. A better estimate of the post-dialysis value is thus the equilibrated value (about 60 min after dialysis hag finished). In practice it may not be necessary to use the equilibrated

value if the magnitude of rebound is small, as is the case in conventional adult dialysis [1]. However in our children the overestimation of Kt/V arising from using the immediate post-dialysis value is about 20%. This may be a clinically important error that necessitates either routinely using the equilibrated post-dialysis sample or making allowance for the error when defining the acceptable range for Kt/V. This work demonstrates that disequilibrium between ICF and ECF exists and that it is of sufficient magnitude to be important in determining dialysis efficiency. It does not invalidate the concept of Kt/V as an index of dialysis efficiency but demonstrates the potential errors that may arise if it is applied uncritically to paediatric dialysis.

A p p e n d i x 1. M a t h e m a t i c a l m o d e l s

In the single-pool UKM urea is removed by dialysis from a volume (V) equivalent to total body water, and the concentration at time (t) on dialysis, Ct, is determined by the equation C t = C o e -K't/V

(1)

where Co = blood urea at start of dialysis, K = dialyser urea clearance and the normalised whole body urea clearance is thus defined by Kt V

- -loge [Cpost/Cpre]

(2)

where Cpost = post-dialysis blood urea and Cpre = predialysis blood urea. This model neglects urea generation, volume change and residual renal function during dialysis. In the two-pool model urea exists in the ICFfECF pools. Dialysis removes urea from the ECF and the flux of urea between the ICF and ECF pools is determined by the mass transfer coefficient for urea, T. Thus the change in ECF urea concentration takes the form dC2 dt

-

1 [T (C1 - C 2 ) - K 9 Ca] V2

(3)

where C1 and C2 are the urea concentration in the ICF and ECF respectively and V2 is the ECF volume. Blood urea represents the ECF concentration. The solution of this equation is given by Ct=A-e-yt+

B.e-zt

(4)

where A, B, y, z are constants which contain the parameters T, K, V1 (the ICF volume) and V2. This model assumes that C1 = C2 at the start of dialysis and ignores urea generation, volume change and renal function during dialysis. The mass transfer coefficient is calculated from the rebound increase in blood urea seen during the 1st h after dialysis when K is zero. The urea concentration in the ECF is given by the solution of Eq. 3 when K = 0. This takes the form C2 (t) = C2 (oo) - [C2 (oo)_ C2 (0)] " e-Xa

(5)

where X = T [I/Vl+l/V2]

(6)

353 a n d C2 (0) = blood urea at end of dialysis, C2 (c~) = equilibrated post-dialysis blood urea. A regression line is fitted (method of least squares) to the log-transformed values for the diference between the actual blood urea and the rebound value at 1 h post dialysis which is assumed to represent the equilibrium value. An estimate of Kt/V can be obtained from Eq. 2 if the equilibrated post-dialysis urea is used instead of the value immediately post dialysis. The volumes of distribution V, V1 and V2 (total body water, ICF and ECF, respectively) are not known, therefore values based on reference data [16] are used (V = 0.6 1/kg body weight, V1 = 0.4 1/kg, V2 = 0.2 1/kg).

References 1. Sargent JA (1983) Control of dialysis by a single pool urea model: the National CooperativeDialysis Study. KidneyInt 23:519-525 2. GotchFA, SargentJA (1985) A mechanistic analysis of the National Cooperative Dialysis Study (NCDS). KidneyInt 28:526-534 3. Buur T, Timpka T, Lundberg M (1990) Urea kinetics and clinical evaluation of the haemodialysis patient. Nephrol Dial Transplant 5: 347 - 351 4. Maur SM, LynchRE (1976) Hemodialysistechniquesfor infants and children. Pediatr Clin North Am 23:843-856 5. Donckerwolcke RA, Chantler C (1987) Dialysis therapy - hemodialysis. In: HollidayMA, Barrett TM, Vernier RL (eds) Pediatric nephrology.Williams and Wilkins Baltimore, pp 799-804

6. GardinerOP, SawyerAN, DonkerwolckeRA, HaycockGB, Murphy A, Ogg C, Winder E, Chantler C (1982) Assessment of dialysis requirements for children on regular haemodialysis. Dial Transplant 11: 754- 757 7. Lopot F (1990) Multicompartment models. In: Lopot F (ed) Urea kinetic modelling. European Dialysis and Transplant Nurses Association - EuropeanRenal Care Association,Geneva, pp 179-181 8. SprengerKBG, KrantzW, Lewis AE, Stadtmuller U (1983) Kinetic modelling of hemodialysis, hemofiltration and hemodiafiltration. Kidney Int 24:143 - 151 9. HarmonWE, SpinozziN, MeyerA, GrupeWE (1981) Use of protein catabolic rate to monitorpediatric hemodialysis. Dial Transplant 10: 324-330 10. SkalskyM, SchindhelmK, Farrell PC (1978) Accuratedetermination of in vivo dialyser clearances. Dial Transplant 7:217 I1. Friis-Hansen B (1961) Body water compartments in children: changes during growth and related changes in body composition. Pediatrics 28: 169-181 12. Leroy D, Dechoux M, Guest G, Broyer M, Sachs C (1985) Extracellular volume and blood pressure in 82 haemodialysed children. Proc Eur Dial Transplant Assoc 22: 847- 850 13. Shaldon S, Deschodt G, Branger B, Granolleras C, Baldamus CA, Lysaght MJ, Dinarello CA (1985) Hemodialysis hypotension, the interleucin hypothesis restated. Proc Eur Dial Transplant Assoc 22: 229-243 14. Lim VS, Flauigan MJ (1989) The effect of interdialytic interval on protein metabolism: evidence suggesting dialysis induced catabolism. Am J Kidney Dis 14: 96-100 15. Pedrini LA, Zereik S, Rasmy S (1988) Causes, kinetics and clinical implications of post haemodialysis urea rebound. Kidney Int 34: 817-824 16. Sargent JA, Gotch FA (1989) Principles and biophysics of dialysis. In: MaherJF (ed) Replacementof renal functionby dialysis. Kiuwer, Durdreckt, p 106

Announcement Rules for the Stiftung C.-E. Alken-Prize 1. We would like to announce the establishment of an endowment fund under the name of Stiftung C.-E. Alken, for the purpose of furthering clinical work and research in the field of Urology by presenting an annual award of SwF. 10,000.- for outstanding scientific endeavours. 2. The award will be for an unpublished work in the field of Urology. An award certificate will also be presented. The prize money may be divided among several contributors. 3. The papers may be written in either German or English, and three copies have to be submitted by October 1, 1992 to Stiftungsrat C.-E. Alken-Stiftung Advokaturbtiro Kellerhals & Partner P.O. Box CH-3000 Bern 7

4. Each submission must be affixed with a code word and must not bear the author's name. But, in addition, a sealed envelope labelled with the code word is to be included, containing on a separate sheet of paper the author's name and professional standing, together with the code word. 5. The adjudicators for the award will be the Board of Directors of the Endowment Fund, and their decision will be final. Bern, April 1992 For the Stiftung C.-E. Alken: The Board of Directors:

Prof. R. Hohenfellner (President) Prof. K. Bandhauer Prof. R. Nagel Prof. E. Zingg