J Am Soc Nephrol 9: 2118-2123, 1998

Whole Body KtIV from Dialysate Urea Measurements during

Hemodialy si s

JAN STERNBYGambro, Lund, Sweden.

Abstract. A new method for the calculation of dialysis dose

from continuous measurements of dialysate urea concentra-

tions has been developed. It is based on urea mass in the patient

instead of plasma concentrations, and results in a measure of

dialysis dose that has been named whole body Kt/V. The

measured urea mass removal rate and the slope of the dialysate

urea concentration curve are the key parameters needed for the

calculations. No assumptions have to be made about urea

distribution in the body (single or double pool, etc.). Blood

sampling is not needed. This simplifies the logistics and elim-

mates the problems with rebound and timing in taking samples.

The total urea mass present in the body before treatment is also

obtained. It can be used directly, or in relation to body weight

or water volume, as a measure of the level of urea in the body.

This may serve as an alternative to pretreatment plasma con-

centration. If a pretreatment plasma urea concentration is avail-

able, the urea distribution volume can be calculated, which

may be of separate clinical interest.

To quantify the amount of dialysis given to a patient, Sargent

and Gotch ( 1 ) recommended using the first-order rate constant

times time, which is the product of dialyzer urea clearance (K)

and treatment time (t), normalized by the urea distribution

volume (V) in the patient, the so-called Kt/V. This quantity has

been found to correlate to mortality. It is the gold standard for

dialysis quantification. For a model in which the whole body

(including the blood) is regarded as a single pool of constant

volume with a homogeneous urea distribution, Kt/V uniquely

determines the reduction in blood urea during treatment and

can be determined by measuring blood urea before and after

the treatment. It is also possible to handle the variations in urea

volume caused by ultrafiltration and the effects of urea gener-

ation during treatment. This is the basis for all current blood-

based methods for dialysis quantification.

However, the blood urea concentration during treatment

often does not fit these first-order dynamics. One possible

explanation for this is internal resistance to urea transport

between different body pools, which will cause a quicker

removal of urea from the blood and/or the extracellular space

than from the rest of the body. This will create a urea concen-

tration gradient between the body pools, which determines the

slower rate of urea transport in steady state. Another explana-

tion is given by the regional blood flow model (2), stating that

some large-volume parts of the body are perfused by relatively

low blood flows, whereas some smaller volume parts are

perfused by relatively high blood flows. This creates a system

of parallel paths with different time constants for the urea

Received February 13, 1998. Accepted May 14, 1998.

Correspondence to Dr. Jan Sternby, Gambro AB, Box 10101, 5-220 10 Lund,

Sweden.

1046-6673/0901 1-21 18$03.00/0

Journal of the American Society of Nephrology

Copyright (0 1998 by the American Society of Nephrology

transport. Whatever model is used, these multiple-body pools

of urea complicate the calculation of dialysis dose.

Another effect of the urea disequilibrium caused by the

treatment is the equilibration that will take place afterward.

The result is a significant posttreatment rise in the blood urea

concentration, the rebound. Although dose calculations based

on blood urea samples may take the effects of urea disequilib-

rium into account, the rebound after treatment makes the

timing important in taking the posttreatment blood sample.

As an alternative to blood side calculations, the spent dia-

lysate can be collected and the total amount of urea removed

can be measured. Normalized for body size, this could be used

as an alternative measure of dialysis dose. The solute removal

index introduced by Keshaviah and Star (3) is similar, but

normalized by the predialysis urea mass. The removed amount

of urea can also be used to calculate Kt/V by direct dialysis

quantification (DDQ) (4), but blood urea values are still

needed, and the effect of urea disequilibrium in the body has to

be considered.

Monitors for the continuous measurement of dialysate urea

concentration have opened new possibilities for accurate dose

quantification in dialysis. Provided that dialyzer clearance is

not changed during treatment, there is a constant relation

between blood and diabysate urea concentrations. Most blood

side formulas depend only on concentration ratios, and can

therefore be used for dialysate side calculations just by replac-

ing the blood side ratios with the corresponding dialysate side

ratios. Again, the effect of urea disequilibrium in the body has

to be considered just as for blood side calculations, which can

be done as described by Smye et a!. (5) or Tattersall et al. (6).A major disadvantage of this method is that clearance has to

stay constant throughout the treatment, otherwise the relation

between blood and dialysate side concentrations is altered.

This prohibits any change in blood flow rate, dialysate flow

rate, or ultrafiltration rate during the treatment.

The two-pool method described and evaluated by Keshaviah

J Am Soc Nephrol 9: 2118-2123, 1998 Whole Body Kt/V 2119

et a!. (7) seems to be derived along these lines, although the

exact mathematical details have not been disclosed in the

literature. With this method, a measurement of the plasma

concentration of urea via an equilibrated dialysate sample is

needed for the calculation of the solute removal index (3) and

the urea mass in the body. There was an excellent agreement

between this method and the dose calculated by a DDQ method

(4) modified to handle the two-pool effect.

The present article describes a new method to quantify

dialysis dose based on diabysate side urea measurements,

which has been named whole body Kt/V (wbKt/%T). The calcu-

lations are not based on concentration ratios. Clearance is

therefore not required to stay constant during the whole treat-

ment, but only for a limited period, long enough to allow the

determination of a slope. In contrast to Keshaviah et al. (7), our

method involves the total mass of urea in the body (mi) instead

of urea concentrations. It is therefore insensitive to the effects

of urea distribution to different body pools, and no compensa-

tion is needed for the urea rebound after treatment.

Materials and MethodsTheory

The mathematical background for the new method is given in theAppendix and in more detail in an article soon to be published (8). To

do calculations on the total mass of urea in the body, we need to defineconcentration and clearance relating to the whole body. A natural

definition of urea concentration at time t is the mean concentration

cmt over an assumed distribution volume V. This volume is not

known, but the definition assures that the total amount of urea in thebody will equal the product of the mean concentration and volume.

Using the mean concentration, we can define a whole body clear-

ance (KWb) as the ratio of urea mass removal rate to mean urea

concentration. None of these quantities can be measured, but we can

still use them for calculations. Our definition of clearance is in

contrast to the common one with the blood urea concentration in the

denominator. The assumed size of the distribution volume will not

affect the ratio KW�,/V (which is needed in the calculation of wbKt/V).

If the actual urea mass is assumed to be distributed in a larger volume,

the mean concentration will be correspondingly decreased. In thedefinition of KWb, the actual urea mass removal rate will then be

divided by this smaller mean concentration, which will give a pro-

portionally increased KWb.

At the start of a treatment, urea is removed mainly from the bloodwith little involvement of the rest of the body. It will take some time

to create the concentration gradients that are responsible for the

transport from tissue to blood. After this initial transient phase, the

steady-state logarithmic slope of the urea concentrations in both blood

and dialysate will be KW�,/V. With the definitions above, it can be

shown (see Appendix) that at each point in time the ratio KWJV equals

the ratio of urea mass removal rate to urea mass left in the body. Theurea mass removal rate is measured on the dialysate side as theproduct of dialysate urea concentration and dialysate flow rate. There-

fore, during parts of the treatment with constant conditions, the total

body mass of urea can be calculated from the urea mass removal rate

and the slope of the dialysate urea concentration curve. Taking urea

generation (G) and accumulated urea removal measured during treat-

ment into account, the pretreatment body mass of urea (m0) can then

be estimated.The total body mass of urea can now be calculated for every

moment of the treatment from the pretreatment mass, accumulated

removal, and urea generation. From the ratio of urea mass removal

rate to remaining total body mass of urea, KW�,/V during the whole

treatment can be found, and from that, by integration over time, thewbKt/V.

In Vitro Tests

The calculation of the pretreatment urea mass mo was tested in six

in vitro treatments. A known amount of urea was added to a container

(23 L) with dialysis fluid (acting as a patient). The container was

subjected to dialysis (2.25 to 3.46 h) with a standard dialyzer at blood

flow rates of 100 to 250 mI/mm and dialysate flow rates of 300, 500,

and 700 mb/mm. The calculated m0 was compared with the known

amount of urea initially added to the container.

Clinical Tests

The new method was evaluated using data from the clinical test of

the DQM 200 urea monitor (Gambro AB) involving 83 treatments atfour centers. This study was approved by the Institutional ReviewBoard/ethics committees at the participating centers, and informed

written consent was obtained from all participating subjects (19 fe-

male and 14 male patients who were studied for two to six treatments

each). The median age was 70 yr (range, 24 to 83), median body

weight 69 kg (47 to 100), and median ultrafiltration volume 2.8 L (0.5

to 6.6). All patients continued their normal dialysis schedules duringthe study, with a median treatment time of 3.45 h (2.45 to 5. 1 ) at a

median blood flow rate of 300 mb/mm (200 to 500). In 13 patientswith residual renal urea clearance (Km), this was 1.5 mb/mm (0.1 to

7.3).

The calculated dialysis dose wbKt/V was compared with blood sideeKt/V (9), which was obtained from the single-pool 1(1/V (Kt/V%�) from

Daugirdas’ second formula (10) by correcting for equilibration.

eKt/V= Kt/V�� x (1 - 0.6/t) + 0.03

Blood samples for the analysis of urea were drawn before and at theend of the treatment. The end treatment sample was drawn from the

arterial line at full blood flow 5 mm before stopping the blood pump,

just prior to drawing a final dialysate sample. Access recirculation was

not measured during the study, but no such clinical signs were seen in

any of the patients at the time of the study or the following 6 mo.

However, a single-digit percentage of access recirculation cannot beruled out for two patients based on regular measurements of total

recirculation. For estimation of urea generation G, the weekly urea

generation was assumed to equal 3 times the removal during the

treatment. For the 13 patients with residual renal urea clearance, thecalculation of wbKt/V was modified to take this into account. The

fraction of urea removal by the native kidneys to removal by dialysiswas assumed to equal the fraction of K� to an estimated clearance of

150 ml/min. Any error in this estimate is relatively insignificant,

because the effect of K� is in itself a small correction.

The calculated pretreatment urea mass mo was used together with

the pretreatment serum concentration to find the urea distribution

volume (V). This volume was compared with the volume calculated

by the Watson formula (1 1), and with the volume calculated by DDQ

(12). The DDQ is based on equating total removal to the product of

volume and concentration change, with ultrafiltration and urea gen-eration taken into account. To get correct results, an equilibrated

posttreatment concentration should be used in the DDQ formula. Thiswas accomplished by finding the posttreatment concentration needed

in the Daugirdas KiN formula (10) to produce the achieved eKt/Vdirectly without any correction. One patient was monitored for more

than a year after the initial study. This was done as a preliminary

J Am Soc Nephrol 9: 2118-2123, 1998

0. 0��

0 � � � 0 0

0 :$�%�#{176}d�;#{176}� 00 00000000 cb

. - - - - � u0-

2120 Journal of the American Society of Nephrology

evaluation of the long-term variation in calculated distribution vol-

ume.

Statistical AnalysisDifferences are given as means ± SEM. Two-sided t test was used

for tests of significance.

ResultsThe calculated initial urea mass m0 and the true value for the

six in vitro tests are shown in Table 1 . The mean difference

was 0.38 ± 0.36 g or 1.2 ± 0.9%.

Figure 1 shows wbKt/V versus eKt/V. Their mean values

(n 83) were both 1.25 ± 0.03, with SD of 0.29 and 0.28,

respectively. The correlation was 0.947 and the difference of

0.00 ± 0.01 was not significant and had an SD of 0.094. The

Bland-Altman (I 3) plot of the difference between the methods

versus their mean value is shown in Figure 2. The 95% limits

of agreement are ±0.18.

The mean volume V was 33.4 ± 0.7 L by the new method,

33.9 ± 0.5 L by Watson’s formula, and 32.8 ± 0.7 L by DDQ,

with SD of 6.2, 4.9, and 6.7 L, respectively. The correlation

between V calculated by the new method and by Watson’s

formula was 0.761 (n = 81) with a nonsignificant difference of

-0.41 ± 0.46 L (Figure 3). The correlation between V calcu-

bated by the new method and by DDQ was 0.912 (n = 83), with

a weakly significant (P = 0.034) mean difference of 0.65 ±

0.30 L (Figure 4). Figure 5 shows repeated determinations of V

by the new method for a single patient during 1 yr.

DiscussionThe possibility of combining the urea mass removal rate and

the slope of the urea concentration curve to find the mass of

urea in the body was confirmed in vitro. The accuracy was

mainly determined by the accuracy in measuring urea removal.

In vivo, ultrafiltration, urea generation, and variability in treat-

ment conditions will make the slope estimation more difficult,

but it should still be possible to keep the error in the urea mass

determination within a few percentage points.

The major advantages compared with other methods for

dose quantification are that no assumptions have been made

regarding the urea distribution in the body and that we only

need a short part of the treatment with steady conditions for the

slope determination. Most of the equations in the derivation

hold exactly without any assumptions. For patients with no

Table 1. Calculated and true initial urea mass in vitro

Test No.Calculated m0

(g)

True m0

(g)

Diffe

(g)

rence

(%)

1 36.23 36.03 0.20 0.55

2 47.76 48.65 -0.89 -1.82

3 37.60 36.04 1.56 4.33

4 37.05 36.05 1.00 2.76

5 35.75 36.04 -0.29 -0.81

6 36.74 36.04 0.70 1.95

wbKt/V

2

1.6

1.2

0.8

1.2 1.6 2 eKt/V

Figure 1. Whole body Kt/V (wbKt/V) versus equilibrated Kt/V (eKt/V)

(Daugirdas) and line of identity.

wbKt/V-eKt/V

0.2

0

-0.2

0.5 1 1.5 2meanKt/V

Figure 2. Bland-Altman plot of the difference between wbKt/V and

eKt/V with the 95% limits of agreement (dashed).

V by new method (L)

50

40

30

20

30 40 Watson V (L)

Figure 3. Urea distribution volume from pretreatment urea mass and

serum concentration versus Watson volume and line of identity.

residual renal function, it is only the estimation of urea gener-

ation and of KWb/V from the slope of the dialysate urea con-

centration curve that require additional assumptions. This is in

contrast to other methods of handling the urea distribution

effects in Kt/V calculations. In the method of Smye et al. (5),

it is assumed that the conditions are constant, so that the rate of

equilibrated urea decrease measured during the final part of the

treatment is representative of the whole treatment. Likewise,

the correction for urea rebound in the calculation of Daugirdas

eKt/V (9) assumes that the treatment efficiency at the end of

treatment equals the mean treatment efficiency for the whole

treatment as measured by single-pool formulas.

In this article, the urea generation has been assumed to be

constant during the whole week. This assumption is not nec-

essary, but facilitates the calculations. The weekly generation

V by new method (L)

Urea volume (L) ,

30 o %��#{176}%#{176}#{176}

20

10

00 100

J Am Soc Nephrol 9: 2118-2123, 1998 Whole Body Kt/V 2121

50

40

30

20

30 40 DDQV(L)

Figure 4. Urea distribution volume from pretreatment urea mass and

serum concentration versus direct dialysis quantification (DDQ) vol-

ume and line of identity.

200 300 days

Figure 5. Urea distribution volume for one patient during 1 yr.

was estimated as three times the urea removal during the

treatment. This is a simplification of the results of Garred (14),

in which the removal during the three treatments of a week was

shown to be 37.9, 32. 1, and 30.0% of the total weekly removal.

Our simplification avoids the need to keep track of the days of

the week and of the urea removal in previous treatments. The

effect of urea generation on the calculated initial urea mass m0is only a few percent. The error caused by our simplified

calculation of urea generation should therefore be far below

1%.

The assumption of a constant urea generation may be ques-

tioned. But if the urea generation varies during the treatment,

there would be a corresponding variation in the estimates of

m0. This was not seen in any of the treatments in the clinical

test. A constant error in the estimation of the urea generation

would give a constant slope in the estimates of m0 This was

also not seen in the clinical test. It is thus concluded that our

simple estimation of urea generation is sufficiently accurate.

For the estimation of KWb/V, we need to find parts of the

treatment with a constant logarithmic slope of the dialysate

urea concentration. If the treatment parameters are not

changed, this can be done after the initial period of 30 to 60

mm, during which the internal gradients of urea within the

body are created. The assumption is that in steady state, the

urea mass removal rate from each part of the body in propor-

tion to the remaining mass of urea in that part is the same.

Access dysfunction and cardiopulmonary recirculation are

known to decrease the blood urea concentration at the dialyzer

inlet. Unless these effects are compensated for, the calculated

dialysis dose will therefore be overestimated when methods

based on concentration ratios are used. One advantage of our

method is that these effects will be automatically included in

the calculations, provided that they stay constant during the

slope determination. This is because the method is based on the

actual removal of urea mass, which will decrease accordingly

in the presence of access dysfunction or cardiopulmonary

recirculation.

The dialysate urea concentration is directly proportional to

the dialyzer clearance. Clotting of the dialyzer leads to a

reduced clearance, and can therefore often be detected as an

excessive decrease of the diabysate urea concentration, which

will no longer show the usual exponential decline during the

treatment. This can be indicated by our method for dose

calculation by a failure to find any constant logarithmic slope

of the diabysate urea concentration for the estimate of KW�,/V.

For treatments with a considerable ultrafiltration, the distri-

bution volume V would be expected to decrease, which would

lead to an increasing KW�,/V if KWh is assumed to be constant.

The logarithm of the diabysate urea concentration would then

normally not be linear in time, and no constant slope would be

found. Instead, it would be possible to do all of the calculations

assuming a linear decrease in the volume V. However, this

problem was not encountered in the treatments in the clinical

test, where it was always possible to find a linear slope of the

logarithm of the urea concentration. A possible reason could be

that KWb may also decrease slightly during the treatment, so

that the ratio KW�,/V is essentially constant.

For simplicity, the mathematical background of the method

was shown only for the case with no residual renal urea

clearance. There are, however, no essential difficulties in re-

moving this assumption. In the clinical test, the fraction of

renal urea removal was assumed to equal the fraction of Kru to

total clearance, for which a value of 1 50 mb/mm was used. The

effect of this correction is in general quite small. For the

median Km of 1 .5 mb/mm it is 1 %, and for one extreme case

with Kru equal to 7.3 mI/mm it is approximately 5%. Even

fairly barge errors in the estimate of clearance used in this

correction will therefore have a negligible effect.

The theory and the clinical tests show that it is indeed

possible to calculate the dose of dialysis given to patients based

on dialysate side measurements of urea concentration without

any manual intervention. This makes the method ideal for

routine dose quantification. The good agreement between

wbKt/V and blood-based eKt/V confirms its usefulness. The SD

of the difference between the two methods (0.094) is in the

same range as the variability in Kt/V determinations based on

blood sampling. In the comparison, it would have been desir-

able to use equilibrated posttreatment blood samples directly

for the blood-based Kt/V instead of the eKt/V according to

Daugirdas. This was not possible in the present study, but will

be done in the next one.

The initial mass of urea in the body is an interesting param-

eter produced by the new method. This could be used as an

alternative to the pretreatment blood urea concentration as a

measure of the bevel of urea in the body, preferably normalized

by body weight or distribution volume.

Together with a pretreatment blood urea value, the estimate

of urea mass has been used to calculate the urea distribution

volume. Long-term changes in this volume despite a constant

body weight might reflect real changes in the proportion of

body water in relation to muscle mass or fat. Other possibilities

are that parts of the distribution volume for some reason do not

always participate in the exchange of urea so that the apparent

distribution volume is decreased, or that the apparent distribu-

tion volume is increased due to urea accumulation somewhere.

It is unclear whether the variations seen over 1 yr in the urea

distribution volume for one patient are due to real changes or

measurement errors. This could be an interesting clinical pa-

rameter to monitor continuously, but more studies are needed

to evaluate its usefulness.

The mean difference between the volume calculated by the

new method and by the Watson formula is small and nonsig-

nificant. The scatter in the data is expected because the Watson

formula is known to give a good average estimate, but with

large variations between individuals. The good agreement with

DDQ volume is expected, because both methods are based on

the same measurement of removed urea. The difference is that

in DDQ the removed urea is related to the difference in plasma

concentrations, whereas in the new method the total amount is

related to the pretreatment plasma concentration. It should be

pointed out that these comparisons are based on pretreatment

volumes. The reason for not using posttreatment volumes is

that our new method gives the pretreatment mass, and that

posttreatment urea concentrations are affected by urea disequi-

librium. Also, the calculation of posttreatment volume from

pretreatment volume and weight loss (or ultrafiltration) is

difficult due to uncertainty about the fraction of water in the

weight loss. A comparison with the gold standard for measure-

ment of urea distribution volumes, i.e. , isotope dilution, is

planned for the next study.

Conclusions

A method for the calculation of a whole body Kt/V has been

derived, which is based on the total mass of urea and is

independent of the urea distribution in the body. The new

method is well suited for routine dose quantification, because

it is based on continuous urea measurements in the spent

dialysate without any manual intervention, and blood sampling

is not required. The Kt/V calculated by the new method has

been shown to correlate well with Daugirdas’ equilibrated

blood side Kt/V.

The new method also allows the calculation of pretreatment

urea mass in the body, and, together with a measurement of the

pretreatment blood urea concentration, the urea distribution

volume. These parameters may prove to be interesting clinical

parameters in their own right, but additional studies are needed

to explore this possibility.

Appendix: Mathematical DerivationDefinitions

t = Time

Cdt Dialysate urea concentration at time t

Qd Dialysate flow rate

U� = Removed urea mass (integral of Qdcd,t) until

time t

= Urea generation

= Total mass of urea in the body at time t

= Initial mass of urea in the body= Urea distribution volume

= Mean urea concentration in the body at time t

= Whole body urea clearance = Urea mass

removal rate/Mean urea concentration

Basic Equations

The urea mass in the body is (by the definition of cmt):

mt=cmtXV (1)

From the definition of KWb, the urea mass removal rate can

be written as:

KWb X cmt Qd X cd.� (2)

Combining equations 1 and 2 gives for the urea mass:

Q d X cd,t

m KW�/V (3)

The urea mass balance suggests that the change in accumu-

bated urea equals the difference between generation and re-

movab:

dm1-�--G-QdXcdl (4)

Calculations

If at any moment KWb/V were known, m� at that moment

could be calculated from equation 3. But for periods with

constant KWb/V (i.e., latter part of treatment, under steady

conditions), m� from equation 3 can be inserted into equation 4

to give:

dcd,l KWb I G \

-d---=_--�--X�cdt_�-) (5)

This is a first-order differential equation with constant co-

efficients. For periods when it is constant, KW�,/V can thus be

estimated by the negative slope of ln(cdl - G/Qd), and m1 can

be calculated from equation 3. Integrating equation 4 with G

assumed constant and using the definition of U�:

m�mo+GXt-U� (6)

With G estimated, e.g. , from weekly urea balance, the initial

m0 can be estimated from equation 6, using all m� values that

could be calculated from equation 3 (constant KWb/V), e.g.,

using the median of all m0 values. Combining equations 3 and

6, the relative efficiency KW�,/V can then be found for the whole

treatment:

KWb Qd X cd.t

Vm0+GXt-U1 (7)

2122 Journal of the American Society of Nephrology J Am Soc Nephrol 9: 21 18-2123, 1998

Integrating this over time gives the dialysis dose wbKt/V.

G

m�

V

cmt

KWb

J Am Soc Nephrol 9: 2118-2123, 1998 Whole Body Kt/V 2123

AcknowledgmentsThe data in this study were provided by S. Lacson and Dr. S.

Bander (Gambro Healthcare, St. Louis, MO), S. Minda and Dr. S.

Schwab (Duke University, Durham, NC), Drs. C. Granolleras and R.

Oul#{233}s(Nimes, France), and Dr. J. Hegbrant (Park Dialys, Lund,

Sweden).

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