Computer Methods and Programs in Biomedicine (2004) 74, 109—128

A novel mathematical method based on ureakinetic modeling for computing the dialysis dose

Manuel Pradoa,*, Laura Roaa, Alfonso Palmab, José Antonio Milánb

a Biomedical Engineering Group, Escuela Superior de Ingenieros, Universidad de Sevilla,Camino de los Descubrimientos s/n, 41092 Seville, Spainb Nephrology Service, H.U. Virgen Macarena, Seville, Spain

Received 17 June 2002 ; received in revised form 5 February 2003; accepted 14 March 2003

KEYWORDSUrea kinetic;Mathematical modeling;Clearance;Kt/V;Urea rebound;Hemodialysis adequacy

Summary A novel normalized single pool urea kinetic model (nspUKM) for the quan-tification of the urea removal, dialyzer urea clearance and urea generation rate duringa dialysis session, is presented. Its major goal is the computation of an accurate es-timate of the fractional dialyzer urea clearance (dKt/V), which is denoted nKt/V,in contrast to the equilibrated Kt/V (eKt/V). This work clarifies the significance ofdKt/V as a complement to eKt/V in hemodialysis (HD) prescription and quantifica-tion. This new model emerges from a generalization of the standard single pool ureakinetic model (spUKM) of the US National Cooperative Dialysis Study (NCDS), iden-tified as gspUKM. Due to their significance, the standard single pool Kt/V (spKt/V)and the eKt/V are also analyzed from gspUKM in this work, with the aim of achievinga better interpretation of the results. Indices nKt/V, eKt/V and spKt/V have beencompared with the dKt/V computed from a published and validated two-pool ureakinetic model (2pUKM). We present the results obtained from a clinical study carriedout on a group of 30 end stage renal disease (ESRD) patients. The limits of agreement(mean± 2S.D. (standard deviation) of the difference) between nKt/V and 2pKt/Vwere − 0.077± 0.72% (percentage of the dKt/V mean), while between eKt/V and2pKt/V were − 13.75± 17.39% and between spKt/V and 2pKt/V were − 1.61± 6.54%.These scores prove that the nspUKM model is able to provide a very accurate estimateof 2pKt/V and thus dKt/V, even with high flux (HF) HD. The presented method joinsthe simplicity of single-pool models to the accuracy of double-pool models, whenthe target is the identification of the dialyzer urea clearance, urea removal and ureageneration rate, although it does not provide a good prediction of the urea dynam-ics. Finally, we think that our analytical and experimental findings throw light on thebehavior and applicability of the different Kt/V indices analyzed.© 2003 Elsevier Ireland Ltd. All rights reserved.

1. Introduction

Up to the 1970s, the probability of failure inhemodialysis (PF) was mainly studied by means

*Corresponding author. Tel.: +34-95-448-7342;fax: +34-637-124-960.

E-mail addresses: mpradov@supercable.es (M. Prado),laura@esi.us.es (L. Roa).

of statistical analysis (SA), without a quantitativeknowledge about the relationships among the pro-tein catabolic rate (PCR), blood urea nitrogen con-centration (BUN) and dialyzer urea clearance kd.This lack of knowledge yielded improper statisticalstudy designs and results about the strong depen-dence between PF and PCR, that may not be valid[1]. The mathematical modeling of dialysis startswith the development of the kinetic of solutes [2].

0169-2607/$ — see front matter © 2003 Elsevier Ireland Ltd. All rights reserved.doi:10.1016/S0169-2607(03)00082-8

110 M. Prado et al.

Subsequently two different kinetic-based criteriaemerged, the first based on the clearance of mate-rials in the range 1000—5000 Da, which may causeneurotoxicity. However, these substances cannotbe measured but only inferred from clinical data[3]. The second criteria was supported by urea andother small measurable solutes [4]. The standard-ization of the latter criteria was achieved after aprospective and multicenter study that began inthe late 1970s (National Cooperative Dialysis Study(NCDS)) [5]. The NCDS results promoted the defini-tion of the fractional dialyzer urea clearance, Kt/V,as a measure of the dialysis dose, where K refers tothe dialyzer urea clearance, t refers to the durationof dialysis and V refers to the urea distribution vol-ume at the end of dialysis. The NCDS demonstrateda step function dependence between PF and Kt/V[1]. The NCDS was based on the single pool ureakinetic model (spUKM) [6], which provided a valu-able tool for quantifying and prescribing dialysis.This index has been denoted as spKt/V, because ofthe mathematical model used for its computation.

The complexity of the spUKM demands compu-tational resources, which are not always availableto physicians. Therefore, much work has been de-veloped to simplify the computation of the spKt/V.Lowrie [7] obtained a very simple method by ignor-ing weight loss rate and intradialysis urea genera-tion. The first [8—10] and second [11] generationformulae of Daugirdas for spKt/V computation werepublished later, achieving a very accurate spKt/V,particularly with the second generation formulae.However, the multicompartmental nature of theurea distribution produces an non-negligible er-ror in the estimate of the urea generation rateand the dialyzer urea clearance computed by thespUKM, compared with the direct dialysis quan-tification (DDQ) method [12,13]. Implications ofthe urea rebound on the reliability of kinetic for-mulae as well as clinical concerns regarding thisnon-homogeneous distribution of urea, started tobe investigated in the 1980s [14—16].

Much work has been done on the study of the ma-jor mechanisms implicated in the urea rebound andabout the design of procedures which overcomethe spUKM limitations. The inter-compartmentalpermeability is the first recognized mechanismable to generate urea gradients into the urea dis-tribution volume. However, the urea rebound isalso a consequence of low perfusion tissues or re-gional blood circulation [17,18]. Although severalarguments considered that the regional blood cir-culation could be a dominant mechanism [19], cur-rently both intercompartmental permeability andregional blood circulation are widely accepted asa cause leading to the urea disequilibrium induced

by hemodialysis (HD) [20,21]. The multi-pool ureakinetic model, and particularly the double-poolurea kinetic model has been largely studied and hasdemonstrated a good accuracy even with signifi-cant urea rebound [22,23], what agrees with theconclusions from Schneditz et al. about compart-mental models predict equal HD doses and volumesthat regional blood flow models [20]. The charac-teristic time of progression of these two urea dis-tribution mechanisms is around 30 min. Two otherfaster processes are involved in the urea rebound:the cardiopulmonary recirculation (CPR) [24] andthe vascular access recirculation (AR) [25,26].

The appearance of high flux (HF) and high ef-ficiency filters has made disequilibrium effectsstronger, compelling the improvement of accu-racy of the Kt/V computing methods. However,multipool kinetic models are more complex thanspUKM, and much more than ‘‘bedside’’ formulae.A trade-off solution led to the definition of theequilibrated Kt/V (eKt/V) [27]. Essentially, thisnew index defined the dialysis dose as a normal-ized measure of the urea removal. The clearanceassociated to the eKt/V is called patient clearanceor whole-body clearance [28]. Index eKt/V can becomputed also from the spUKM, using a BUN sampleof the patient at 30—60 min after the end of dialy-sis (equilibrated BUN) as a mathematical boundarycondition (bc) for the BUN at the end of HD, withinthe model. Therefore, the computational complex-ity required for the calculation of eKt/V is equal tothat of the spKt/V, despite a new blood sample isrequired. To solve this latter problem Smye et al.developed a formula to estimate the equilibratedBUN from an intradialytic measurement [29,30].Later on Daugirdas and Schneditz proposed anotherformula, called ‘‘rate equation’’, to compute anestimate of eKt/V (eKt/Vest) which does not requireany additional measurement [19]. Many clinicalworks have demonstrated the accuracy of the rateequation, and the inferior behavior of the Smye andsimilar intradialytic-based methods. Some currentpublished works regarding those results are citedin [31—34]. It must be emphasized that eKt/V isnumerically and conceptually different to dialyzerKt/V. The latter is denoted here as dKt/V to re-mark this issue, and to indicate its difference withrespect to spKt/V, due to spUKM assumptions.

The prospective NCDS has serious limitations withthe non-homogenous solute distribution. Taking intoaccount the evolution of dialyzers, the aging of theend stage renal disease (ESRD) population and thelack of knowledge about clinical outcomes when ahigh dialysis dose (Kt/V) is applied, a new largeprospective multicenter study (HEMO) was designedto advance in those issues. Begun in 1995 [35] with

A novel mathematical method based on urea kinetic modeling for computing the dialysis dose 111

a study design called for 1.5 years of initial recruit-ment and 5 years of further follow-up, the HEMOstudy uses the eKt/V as a dialysis dose index, be-cause of its ability to overcome the urea disequilib-rium induced by HD [36].

Although there are recent works claiming that therelationship between PF and Kt/V may also be dis-covered within other dialysis dose variables as theurea reduction rate URR1 [37,38], the importanceof the Kt/V as a quantitative normalized index forthe adequation of the HD is widely recognized [39].This is confirmed by the selection of Kt/V as a pre-ferred index to quantify HD, within the Dialysis Out-come Quality Initiative (DOQI) Guidelines, from theNational Kidney Foundation (NKF) [40,41].

Several problems arise from this situation. Firstof all and from a clinical point of view, the nephrolo-gist lacks of a quantitative procedure that connectsthe patient urea removal dialysis dose to be pre-scribed, e.g. eKt/V, to the required dialyzer clear-ance. As a second issue, although eKt/V seems avery good measure of urea removal, because it doesnot depend on urea rebound, the lack of correlationbetween BUN levels and patient outcomes indicatesthat urea has not concentration-dependent toxicityand that generation rate of putative toxic low MWsolutes is not proportional to urea generation [39].Lack of knowledge about toxins and thus kineticand mechanisms that originate this behavior, sug-gest that may be important to know both eKt/V anddKt/V in subsequent research studies about thisissue. Although dKt/V could be approximated byspKt/V, which in turn can be estimated adding therate constant to the eKt/V, as the in rate equation[38], this is an inaccurate procedure because of thedeviation between spKt/V and dKt/V [38,42]. Anaccuratemeasure of dKt/V can be performed from atwo-pool urea kinetic model (2pUKM). However, thesimplicity of spUKM and related bedside formulaewould then be lost, requiring both more computa-tional resources, and what is even more important,more physiological data, as intercompartmentalpermeability and compartmental volumes.

We present the novel normalized single pool ureakinetic model (nspUKM), derived from the stan-dard spUKM [6], with the objective of computingan accurate measure of dKt/V. The mathematicaldevelopment of the nspUKM has been describedas an evolution of the standard spUKM, by meansof a generalization of this latter model (gspUKM).The spKt/V and eKt/V, together with the urea gen-eration and urea removal are also calculated and

1 URR is defined as 100 ·(BUN predialysis−BUN postdialy-sis)/BUN predialysis.

analyzed from the gspUKM. We have followed thismethodology to clarify the benefits of these latterindices and this way helping to find a consensusabout ‘‘the most adequate index’’, according tothe goal of several authors [43—45].

We carried out a clinical study with 30 ESRD pa-tients with the purpose of validating the nspUKMmodel. The reference value for dKt/V was from a2pUKM used in [23]. We have called 2pKt/V to thedKt/V estimate. The ability of 2pUKM to predicturea concentration evolution has been validated inseveral studies, particularly in [23], where it wascontrasted by means of a blood side (ultrafiltrated)continuous on-line urea monitoring system (UMS).The proper use of this model under the contextpresented in this paper is commented in Section 3.The spKt/V and eKt/V indices were also computedand compared with the reference 2pKt/V. Devia-tions of these indices were contrasted to those ofthe nspUKM Kt/V index (nKt/V), confirming the an-alytical results obtained in the mathematical analy-sis. We also present a second implementation of theformal nspUKM that uses a prediction of the equili-brated BUN, thus avoiding the third blood drawingto the patient. Both its Kt/V index (nKt/Vest) and theestimate of the equilibrated BUN, have also beenvalidated. Finally, the accuracy of the urea gener-ation computed by the different analyzed methodshas also been tested and presented in this work.

2. Normalized single pool ureakinetic model

2.1. Generalized single pool urea kineticmodel (gspUKM)

We are interested in a single pool model able toyield an accurate estimate of urea nitrogen genera-tion rate, urea removal and dialyzer urea clearance,kd, from standard BUN samples, which are drawnbefore the beginning of HD, between 30 s and 2 minafter the end of HD, and 30—60 min afterwards.With that objective, and considering that the ureanitrogen removal is equal to k ·AUC, where AUC isthe area under the curve of urea nitrogen concen-tration, c(t), during the HD session, it is possible toadjust k, keeping urea removal constant, by meansof the AUC modulation. The latter is achieved bythe adjusting of the model dialysis session duration,TSP. This is the key idea that underlies the general-ized single-pool model (gspUKM).

According to its definition, the parameter TSP

does not aim to achieve any physical meaning. Inthis manner, gspUKM will be state in such a way thatwhen TSP will be equal to the true session duration,

112 M. Prado et al.

T, then gspUKM will match the standard spUKM [6].On the contrary, when TSP will be adjusted with thegoal that predicted k equals kd, then gspUKM willmatch nspUKM. Unlike previous standard spUKM,the novel nspUKM does not aim at the prediction ofthe urea dynamics, as it is clarified below.

2.1.1. Mathematical formulationThe urea kinetics of a well-stirred single-pool isgiven by the following well-known system:

dVdt

= β

Vdcdt

= G − kc − βc (1)

where G is the urea nitrogen generation rate andk is the dialyzer urea clearance, which makes zeroduring interdialysis. Defining Q UF as the ultrafiltra-tion flow rate and θ as the interdialysis period du-ration, then β= −Q UF , during HD session, whereasβ=Q UFT/θ, during interdialysis period. The scaleT/θ ensures that dialysis weight loss will be recov-ered during interdialysis period. This is an acceptedtechnical hypothesis of the standard spUKM [6].

The generalized model is deduced from (1), con-sidering an adjusted model HD duration, TSP, differ-ent from T, as commented previously. The gspUKMwill be given by the following system:

dVdt

= β

Vdcdt

= G − kc − βc, (2)

where the parameters are expressed now aspiece-wise step functions, whose values, forn linked HD sessions, vary among intra- andinter-dialysis periods in the following way:

l ∈ [tl + T − TSP, tl + T] (3a)

k = k, β = −QUFTTSP

, G = GTTSP

(3b)

t ∈ [tl + T , tk+1] (3c)

k = 0, β = QUFTθl, G = G, (3d)

being tl the predialysis instant of session number l(for l= 0, . . . , n− 1), and θl is the subsequent in-terdialysis interval duration. The temporal inter-val (3a) pertains to the dialysis session number l,while the temporal interval (3c) pertains to the sub-sequent interdialysis period. The piece-wise stepfunctions (3) and the ODE system (2) are not definedin [tl, tl +T−TSP], because both c(t) and V(t) arenot modified within this interval. This will be for-malized later by the connection conditions (cc) (5).

Finally, the scale factor T/TSP assures that the totalurea nitrogen generation (G ·T), and weight losses(Q UF ·T) during an HD session will be not affectedby TSP.

Denoting by cv(t) the vascular access urea nitro-gen concentration, at instant t, the boundary con-ditions (bc) for gspUKM are as follows:

c(T − TSP) = cv (0)

c(T) = cv (T + τ)VT + δV(τ)

VT− Gτ

VTc(p) = cv (p)

V(T) = VT (4)

The first bc of (4) specifies the urea nitrogen con-centration at the beginning of the HD. Since c(t)is constant during t∈[0, T−TSP], then c(t)= cv(0)within this interval. The second bc is equivalent toc(T+ τ)= cv(T+ τ), being δV(τ) the weight gain dur-ing the τ units of time after the end of dialysis.Nevertheless, we use the slightly more complicatedequation into (4), because it is usual the definitionof c(T) as bc in current single-pool models. The thirdbc specifies the urea nitrogen concentration at theend of the assessment period, p. Finally, the post-dialysis value VT has been chosen as a bc for theurea distribution volume, as usual.

Finally, cc (5) link the interdialysis l− 1 with theHD session l, as follows:

c(tl + T − TSP) = c(tl)

V(tl + T − TSP) = V(tl) (5)

2.1.2. Solving gspUKMIf bc (4) are known, then gspUKM can be solvedto provide two parameters, the same way thanthe standard spUKM [6]. Because Q UF and T arewell-known parameters, we are interested only inVT , G and k. Although the standard spUKM uses k asan input parameter, obtaining VT and G, we havechosen VT as an input parameter in this work, asshown in Fig. 1. This choice simplifies the use ofthe gspUKM to prescribe a dialyzer urea clearancekd for an HD session. Indeed, this use is not a novelissue of this work, since for example Bankhead

Fig. 1 bc and parameters used for the gspUKM. T, Q UF

and TSP are always input parameters while G, k and VT

may be interchanged (see text).

A novel mathematical method based on urea kinetic modeling for computing the dialysis dose 113

et al. [46] proceed this way in a comparative studyfor the evaluation of the accuracy of the urearemoval estimate, among several mathematicalmethods.

The VT parameter may be obtained from an an-thropometrical formulae or even by kinetic mod-eling. The computation of the urea distributionvolume through the spUKM in a first control dial-ysis session is a classical technique [6]. Currently,this technique can be optimized by means of animproved multi-pool kinetic model or even by theDDQ method. With regard to the computation ofVT by anthropometrical methods, the Hume andWeyers regression [47] or the Watson et al. regres-sion [48] are commonly used formulae. Moreover,an ESRD population-specific formula has recentlybeen proposed by Chertow et al. [49].

The Kt/V index is computed subsequently asKt/VT , according to the nomenclature of themodel.It is remarked that the index is computed using thetrue T, although TSP affects to the computation ofk.

At last, both the classical three-buns method [6]and the two-buns method [50], may be used tosolve the model. These methods are related to thebc cv(p). When the three-buns method is selected,cv(p) is the vascular access urea nitrogen concen-tration at the beginning of the subsequent dialysissession, i.e. p=T+ θ. However, a periodic behav-ior of cv(t) is considered for the standard two-bunsmethod, assuming that cv(p)= cv(0), where p is theperiod. Last method is frequently used with stableESRD patients.

2.2. Definition and analysis of nspUKM

The following analysis is performed for p=T+ θ.Nevertheless, the analytical results may be gener-alized for an assessment interval p>T+ θ, i.e. formore than one dialysis session.

The urea disequilibrium induced by HD disappearsabout 30 min after the end of dialysis. This time in-terval is called equilibrium interval (tequ) [28]. In thesubsequent analysis we will consider that the trueweight gain is constant both during dialysis intervaland during interdialysis interval. This assumptioncan be accepted if we are interested in average pa-rameters. Indeed, it is a commonly used assumptionsince Sargent’s work [6].

The true net urea nitrogen loss Ptu between t= 0and t=T+ tequ is Ptu = (VT +UF)cV (0)− (VT + δV)cv(T+ tequ), where δV is the weight gained dur-ing the equilibrium interval, tequ, and UF is theweight loss during the HD session. The single poolnet urea nitrogen loss at the same interval isPu = (VT +UF)c(0)− (VT + δV)c(T+ tequ). Because of

the first bc in (4) the difference between bothequations can be written as follows:

Pu − Ptu = (VT + δV)[cv (T + tequ)− c(T + tequ)] (6)

Since urea is homogeneously distributed afterT+ tequ, if τ= tequ then c(T+ tequ)= cv(T+ tequ) and,therefore, Pu − Ptu is null. However, if τ= 0, thenthe difference (Pu − Ptu)/(VT + δV) is equal to theurea rebound cv(T+ tequ)− cv(T). As a consequence,we deduce that the difference Pu − Ptu is producedby the urea rebound, and can be compensated by aproper selection of the delay, τ, between the endof HD and the blood drawing instant. ConsideringGt

h as the true average urea nitrogen generationrate in the interval where urea is homogeneouslydistributed, i.e. [T+ tequ, p], and using (6) togetherwith the third bc in (4), the urea nitrogen bal-ance in [T+ tequ, p] yields the following equationfor G:

(p − (T + tequ))(G − Gth)= Pu − Ptu

⇒ G=Gth + Pu − Ptu

(p − (T + tequ))(7)

Considering now Gtnh as the true average urea ni-

trogen rate in the interval [0, T+ tequ], togetherwith the first bc in (4) and (7), the urea nitrogenbalance in [0, T+ tequ] yields an equation that ac-counts for the urea removal deviation:∫ T

T−TSPkc(t)dt︸ ︷︷ ︸

sp urea removal

−∫ T

0kdcv (t)dt︸ ︷︷ ︸

patient urea removal

= (T + tequ)(G − Gtnh)+ Pu − Ptu

= (T + tequ)(Gth − Gt

nh)+p(Pu − Ptu)

p − (T + tequ)(8)

According to the previous definition of AUC, thefirst integral in (8) is denoted as k ·AUC(T−TSP,T), and the second as kd ·AUCv(0, T). The differ-ence AUC(T−TSP, T)−AUCv(0, T) is related to theurea inbound (Appendix A). Therefore, from (7)and (8), follows that the deviations between G andGth and between k and kd are due to the urea re-bound by Pu − Ptu, together with the urea inboundby AUC(T−TSP, T)−AUCv(0, T). These deviationscan be minimized if τ=tequ and TSP is defined sothat AUC(T−TSP, T)=AUCv(0, T). Last equation iscalled areas condition. The last assignments de-fine the nspUKM as a particular case of gspUKM.Fig. 2 shows c(t) against cv(t) during an HD sessionand the subsequent interdialysis interval for thismodel.

By imposing these conditions, the deviations be-tween estimated nspUKM parameters and patient

114 M. Prado et al.

Fig. 2 Temporal evolution of cv (t) (dotted line) andc(t) (solid line) for the nspUKM, indicating the bc assign-ments.

average parameters, are left as:

k − kd = (T + tequ)(Gth − Gt

nh)

AUCv (0, T)(9a)

G − Gth = 0 (9b)

Equations (9) show that any difference betweenGt

h and Gtnh will originate an error in the estimated

clearance, k. Moreover G is indeed an estimate ofGt

h. These are interesting outcomes, because sev-eral experimental studies conclude that an increasein urea nitrogen generation rate is induced duringintermittent therapy of uremia [51].

How the areas condition is implemented andsolved to obtain TSP is an external issue to the for-mal nspUKM. However, it affects the accuracy ofthe kd estimate. We present an implementation ofnspUKM in this work. This provides an explicit for-mula to calculate TSP. That formula was validatedin a clinical study that is presented below.

3. Methods and materials

The following subsections describe the clinicalstudy design and the computational proceduresdeveloped to validate an implementation of thenspUKM.

3.1. Patient characteristics and clinicalprocedures

The study was carried out with 30 stable ESRDpatients, receiving three HD sessions per week(Monday, Wednesday and Friday). All patients wereanuric. Arteriovenous fistula was used for the vas-cular access in all of them. AR was smaller than10% in 20 of them. Nevertheless, the AR was lim-ited in the remaining patients, who had adequateBUN levels. AR measurement was done by thetwo-needle urea-based method [52], founded onthe formula (P−A)/(P−V), where A and B are BUNsamples drawn from the arterial and venous lines,respectively, and P is a BUN sample drawn from the

Table 1 Major clinical and anthropometrical patientparameters

Parameter Mean± S.D. Range

Age (years) 56± 14.05 22—71Body weight (kg) 68.47± 15.68 43—123Body weight loss

(kg)2.24± 1.096 0.5—4.6

Blood flow(ml/min)

261.17± 24.76 210—300

Treatmenttime (h)

4.03± 0.29 3—4.5

BUN1 (mg/dl) 69.38± 23.93 17.18—123.36BUN2 (mg/dl) 22.16± 9.14 7.45—40.65BUN3 (mg/dl) 25.95± 10.65 8.41—47.66Sex, male/female 12/18

arterial line 2-min after switching the dialyzer tobypass with a reduced blood flow.

Three BUN samples were taken, prior to HD(BUN1), at the end of treatment (BUN2), and 30 minafterwards (BUN3). Hence, the equilibrium intervalwas taken as tequ = 30 min. The BUN samples weremeasured by a standard analyzer method. Post-dialysis BUN samples were taken 1-min after theend of the HD, keeping the arterial line blood flowequal to 50 ml/min. Dialysis operating conditionswere maintained constant during the HD session.Dialysate flow rate was 500 ml/min for all patientsand sessions. The main clinical and anthropomet-rical parameters, indicated as mean± standarddeviation (S.D.), are summarized in Table 1.

3.2. Two-pool model

A two variable volume urea kinetic model (2pUKM)has been used to measure dKt/V. This model, whichis described in Appendix D, has been validated byseveral authors. Canaud et al. confirmed the highaccuracy of the urea concentration predicted bythis model, with measures taken at 1-min inter-vals by a continuous on-line UMS, and verified thaturea rebound fitted the model prediction perfectly[23]. In that referred work, authors estimated thepredialysis urea concentration, the intercompart-mental mass-transfer coefficient, Kc, the fractionof volume of accessible compartment, fv1, themass-transfer coefficient that characterized theHD filter, and the postdialysis urea distributionvolume, VT , by fitting of all data collected for anindividual during different dialysis systems. TheVT estimate was assumed constant over all dialysissessions delivered to a patient. Moreover, authorsconsidered a fixed urea generation rate of 10mg/min in all patients and sessions. Although we

A novel mathematical method based on urea kinetic modeling for computing the dialysis dose 115

do not agree with the assumption of a fixed G forall patients, however, this issue does not affect tothe ability of 2pUKM to predict accurately the ureaconcentration, what was confirmed in the afore-mentioned study. We can expect that this accuracywill be preserved if 2pUKM is implemented withthe same degrees of freedom than in [23]. Withthat goal, we estimated fv1, Kc and kd, for eachpatient and session by fitting the model accessibleurea concentration, c1, to the measured urea ni-trogen concentration at predialysis, c1(0)=BUN1,postdialysis, c1(T)=BUN2, and 30-min after theHD, c1(T+ tequ)=BUN3, and finally assuring thaturea reaches the equilibrium at tf =T+ tequ. Lat-ter was forced making c1(tf)= c2(tf), being c2(t)the non-accesible urea nitrogen concentration.The sensitivity of 2pKt/V to the variation oftf (keeping the remaining conditions) was verylow, and so we can neglect small deviationson tf.

The total urea distribution volume at the end ofdialysis, VT , needed as input to 2pUKM, was com-puted by the anthropometrical formula of Chertowet al. [49], which has been written in Appendix Gfor convenience. It has been selected because pre-vious studies have confirmed a better accuracythan that obtained by Watson et al. [48], and Humeand Weyers formulae [47]. We verified again thevery low sensitivity of the 2pKt/V to the deviationof VT .

Finally, an accuracy estimate of the urea ni-trogen generation rate, which is also an inputto this model, was computed by the espUKM, asdescribed in Appendix B.2. The accuracy of thiscomputation has been confirmed in several clin-ical studies [46]. The espUKM was solved by thetwo-buns method as described in Appendix F. Fi-nally, Appendix D shows also a variant of the afore-mentioned two pool model. This was run to verifythat urea carried by water has very low influenceon 2pKt/V.

3.3. Single pool models

The spUKM and espUKM (Appendix B), together withthe novel nspUKM, were run to provide the true ureanitrogen generation rate, Gt, and dKt/V estimates.All of them were solved by the two BUN technique.Appendices E and F describe both the analyticalsolution and the computational procedure used tosolve them.

An analytical study was performed to obtain anexplicit formula for the TSP parameter. This was ob-tained by an approximation for cv(t) based on thethree BUN samples aforementioned. This study hasbeen briefly described in Appendix C.

Finally, we include in this work the results ob-tained by the nspUKM when BUN3 is not measured,but estimated. Appendix E presents details aboutthe last computational method.

3.4. Statistics

We have used paired Student’s t-tests to comparethe Kt/V computed by single pool models with the2pKt/V taken as reference (dKt/V estimate). Re-gression coefficients have not been computed forassessing agreement between two methods and in-stead the limits of agreement (mean± 2 S.D. of thedifference) suggested by Bland and Altman [53] to-gether with scatter plots showing the differenceversus the mean are presented. The 95% confidenceinterval both for mean and for 2S.D. of the differ-ence, is also presented. The standard skewness andkurtosis parameters are used to measure the devi-ation of the paired differences from a normal dis-tribution. The statistical significance of the null hy-pothesis: mean of the difference equal to zero, isalso provided. An α= 0.05 was taken as level of sta-tistical significance.

Urea nitrogen generation rates are comparedby the same procedure. Other sample values aredescribed as mean± S.D. Box-and-whisker plots ofsome interesting values are also included.

An explicit formula for TSP was developed froma phenomenological analysis which in turn re-quired a linear regression for an internal coef-ficient. The statistical details are described inAppendix C.

4. Results

4.1. Two-pool model

Intercompartmental urea mass transfer coeffi-cient Kc, obtained solving 2pUKM as indicated inSection 3was 561.7± 142.4 ml/min. Mean is lowerthan 912 ml/min, which was obtained by Canaudet al. [23]. Urea distribution volume obtainedby Chertow formula was 36.56± 8.19 l, which is53.89± 6.89% of dry-weight. This value is closeto other published data. Ratio between accessi-ble and non-accessible compartments (V1/V2) was0.6± 0.23. This is a value closer to 0.5, which is aparadigm for the ratio between extracellular andintracellular volume, than the value 0.31 obtainedby Canaud et al. in the aforementioned study.

The 2pKt/V computed by 2pUKM was 1.42± 0.26.A box-and-whisker plot for this value is presentedin Fig. 3.

116 M. Prado et al.

Fig. 3 Box-and-Whisker plot of Kt/V indices obtained in the clinical study. 2pKt/V is the dKt/V estimate used asreference, as indicated in text. Remaining indices are clarified in text.

4.2. Single-pool models

Kt/V indices were computed by the new nspUKM andby the standard spUKM [6] and espUKM. Their imple-mentation is described in Appendix F. The nspUKMimplementation accounted for two Kt/V indices.The first of them (nKt/V) is provided when the threeaforementioned BUN samples are known, while thesecond (nKt/Vest) is based on a computational pro-cedure for estimating BUN3. Both are described inAppendix E.

Equation (C.2), which is needed to compute Tsp

by (C.3) and (C.4), was obtained for a validationgroup of 15 patients randomly selected from thewhole study group. Measured scale factor, am(R,Re), was obtained solving (C.3) for a Tsp such thatnKt/V was equal to 2pKt/V (dKt/V estimate). Thescale factor function a(R, Re) and its arguments, Rand Re, are defined in Appendix C. Results appearin Table 2. Standard error (S.E.) was 0.0034 andR2 = 99.83%. As indicated in the aforementionedtable, the equation of the fitted model was a(R,Re)= − 0.9645 ln(Re) + 0.8939 ln(R) + 0.9811. TheANOVA test P-value was P< 0.00005 and thus therewas a statistically significant relationship between

Table 2 Linear regression equation for predicting the scale factor function a(R, Re)

Target group Predictor variables Regression coefficient S.E. R2 RMSE (%) P-value

Validation 0.0034 0.998 0.3631 0.0000β1 −0.9645 0.0149 0.0000β2 0.8939 0.0112 0.0000β3 0.9811 0.0061 0.0000

Cross validation 0.5731Total 0.0047 0.994 0.5215 0.0000

β1 −0.9664 0.0154 0.0000β2 0.8984 0.0129 0.0000β3 0.9825 0.0058 0.0000

Regression coefficients, standard error (S.E.), R2 and P-values corresponding to the ANOVA test (Validation andtotal rows) and to the hypothesis test (mean= 0) for each independent variable.

the variables at the 99.995% confidence level. Fur-thermore, P< 0.00005 for all independent variables(null hypothesis is that the parameter is equal to0), and consequently all independent variables havestatistical significance.

This model was afterwards applied to a cross-validation group formed by the remaining 15 pa-tients. The root mean square error (RMSE) for thevalidation group was 0.3631% (percentage respectto am), and for the cross-validation group was0.5731%. The last value is very close to the previousone, and therefore, we concluded that this linearregression for a(R, Re) has a high performance. Thefinal prediction equation was developed as a linearregression for the total of patients of this study.Table 2 presents the resultant model together withthe statistical significance. Final equation was leftas follows:

a(R, Re)= − 0.9664 ln(Re)+ 0.8984 ln(R)+ 0.9825

(10)

There were two studentized residuals greaterthan 2.0, both are also influential points becauseof their large DFITS. A third influential point had a

A novel mathematical method based on urea kinetic modeling for computing the dialysis dose 117

Fig. 4 Scale factor a(R, Re) predicted by equation (10)vs. the measured value am (see text).

leverage more than five times the average lever-age. All of them have been kept.

Fig. 4 shows the relationship between predicteda(R, Re) and measured am(R, Re). Fig. 5 shows theblood urea concentration approximation cvest(t)for a randomly selected patient from the study,against the cv(t) predicted by the 2pUKM. Despitethis is a gross approximation of cv(t), we obtaineda high accurate implementation of the areas con-dition. Fig. 6 illustrates the decrease of TSP/T ver-sus Re, when T= 4 h, tequ = 30 min and R= 0.248(URR= 75.2%). This value of URR is slightly higherthan 70%, which is the minimum value recom-mended by the NKF for HD schemes of three ses-sions per week.

A box-and-whisker plot for all Kt/V indices ispresented in Fig. 3. Box heights span from lower toupper quartiles (interquartile range). No suspectoutliers have been found. Horizontal line (me-dian) is very close to mark point (mean) in all in-dices. Both interquartile range, mean and whiskers(smallest/highest point within 1.5 interquartile

Fig. 5 BUN concentration approximation, cv est(t), (dot-ted line) for a randomly selected patient from the study,against the cv (t) predicted by the 2pUKM (solid line near-est to dotted line). The non-accessible urea concentra-tion has been also presented (upper solid line).

Fig. 6 Relationship between Tsp/T and Re, for R= 0.248(URR= 75.2%), tequ = 30 min and T= 240 min.

ranges from the lower/upper quartile) of nKt/Vand nKt/Vest are very close to those of 2pKt/V.However, interquartile and whiskers are reducedin eKt/V and spKt/V. Mean eKt/V is near 0.2 unitslower than mean 2pKt/V.

Limits of agreement, 95% confidence intervals ofmean and 2 S.D., and hypothesis test (mean= 0)of the difference between single-pool model Kt/Vindices and 2pKt/V are presented in Table 3. ThenKt/V index predicted 2pKt/V very accurately, ac-cording to its limits of agreement (− 0.077± 0.72%,P= 0.26). The spKt/V index slightly underestimated2pKt/V (− 1.61± 6.54%, P< 0.05). The eKt/Vstrongly undervalued 2pKt/V (− 13.75± 17.39%,P= 1.5× 10− 9). The last result confirms the signif-icance of the urea inbound due to urea gradientsthat appear during HD session. This way, althougheKt/V is an adequate index to quantitate dialysisdose, it cannot be used to predict the dialyzerurea clearance. Finally, nKt/Vest also predictedaccurately enough 2pKt/V (0.35± 3.10%, P= 0.24).

Bland—Altman plot comparing nKt/V against2pKt/V is presented in Fig. 7. Despite the fact thatthere is a slight increase of deviations with Kt/V

Fig. 7 Bland—Altman plot comparing nKt/V to 2pKt/V.Limits of agreement are represented by the dotted line(mean of difference) and the two solid lines (± 2 S.D. ofdifference).

118M.Prado

etal.

Table 3 Limits of agreement (mean± 2 S.D.) between the different methods tested and the reference 2pKt/V (last row compares the eKt/Vest from rate equationwith the standard eKt/V)

Paired difference Units of agreement 95% Mean confidenceinterval

95% 2S.D. confidenceinterval

P-value Standard skewness Standard Kurtosis

nKt/V− 2pKl/V − 0.0011± 0.0102 − 0.0011± 0.0019 [0.0080—0.0136] 0.26 0.67 0.69(− 0.077± 0.72)% (− 0.077± 0.134)% [0.56—0.96]%

SpKt/V− 2pKt/V − 0.0228± 0.0928 − 0.0228± 0.0173 [0.0739—0.1248] 0.01 − 4.89 6.83(− 1.61± 6.54)% (− 1.61± 1.22)% [5.20—8.79]%

eKt/V− 2pKt/V − 0.1953± 0.2469 − 0.1953± 0.0461 [0.1966—0.3319] 1.5× 10− 9 − 5.55 9.04(− 13.75± 17.39)% (− 13.75± 3.25)% [13.85—23.37]%

nKt/Vest − 2pKt/V 0.0049± 0.0440 0.0049± 0.0082 [0.0350—0.0591] 0.24 1.12 1.70(0.35± 3.10)% (0.35± 0.58)% [2.46—4.16]%

eKt/Vest − eKt/V − 0.0140± 0.1303 − 0.0140± 0.0243 [0.0519—0.0876] 0.25 4.15 6.49(− 1.138± 10.593)% (− 1.138± 1.976)% [4.220—7.12]%

Percentages are referred to the mean 2pKt/V= 1.42 (mean eKt/V= 1.23 for the last row). The 95% confidence intervals for mean and 2S.D. are also presentedto show the accuracy of the limits of agreement. The absolute value of standard skewness and kurtosis statistics must be lower than 2 for a normal distribution.

A novel mathematical method based on urea kinetic modeling for computing the dialysis dose 119

Fig. 8 Bland—Altman plot comparing eKt/V obtained byespUKM to 2pKt/V. Limits of agreement are representedby the dotted line (mean of difference) and the two solidlines (± 2 S.D. of difference).

mean, the limits of agreement properly representthe accuracy of this method in all the range ofstudy. Bland—Altman plot comparing eKt/V against2pKt/V (Fig. 8) confirmed that its undervaluationincreases with the Kt/V mean, as expected. Thisbehavior proves that urea inbound, given by the in-equality (A.1), increases with dialysis dose. Never-theless, this relationship will be modulated by dial-ysis duration T. Bland—Altman plot of spKt/V versus2pKt/V (Fig. 9) shows that its undervaluation growswith the mean Kt/V, starting from a value near 1.2.This fact indicates that the urea inbound is the dom-inant term in (B.2) in opposition to the urea re-bound term. Under these conditions (B.1) predictsthat spKt/V<dKt/V. Finally Bland—Altman plot ofnKt/Vest versus 2pKt/V (Fig. 10) confirms the accu-racy of this index, despite the fact that its meandeviation grows with mean Kt/V. Lack of accuracyof nKt/Vest regarding nKt/V is related to the preci-sion of the estimated BUN3 (BUN3est), which in turndepends on the rate equation.

Table 4 shows the deviation of BUN3est againstBUN3, while Table 3 shows the deviation betweeneKt/Vest and eKt/V. In addition, Bland—Altman plot

Fig. 9 Bland—Altman plot comparing spKt/V obtainedby the standard spUKM to 2pKt/V. Limits of agreementare represented by the dotted line (mean of difference)and the two solid lines (± 2S.D. of difference).

Fig. 10 Bland—Altman plot comparing nKt/V obtainedby nspUKM when the BUN3 is not known to 2pKt/V. Limitsof agreement are represented by the dotted line (meanof difference) and the two solid lines (±2S.D. of differ-ence).

of eKt/Vest against standard eKt/V is presented inFig. 11. According to these results, BUN3est overval-ued BUN3 (6.24± 10.10%, P< 10− 6). Percentagewas taken with respect to the measured meanBUN3 = 25.95 mg/dl. The accuracy of the rateequation [19] to estimate the standard eKt/V whenBUN3 is unknown was confirmed (− 1.138± 10.59%,P= 0.25). However, it had a significant dispersion.

The accuracy of urea removal and urea gener-ation rate G computed by espUKM has been con-firmed in many works [46,23]. Therefore, we usedthat G as a reference in this study. Results areshown in Table 4. Urea generation rate computedby nspUKM was equal to that computed by espUKM,as was expected according to (9) and (B.3b), andthis is the reason why nspUKM does not appearin the aforementioned table. Taking percentageswith respect to the reference mean urea nitro-gen generation rate G= 5.85 mg/ml, the limitsof agreement of the Gsp obtained by the standardspUKM was 8.27± 8.87% (P< 10− 10). As was ex-pected by (8) and (7), standard spUKM significantlyovervalues urea generation rate and urea removal.

Fig. 11 Bland—Altman plot comparing the eKt/V com-puted by espUKM to the eKt/Vest calculated by the rateequation (C.2) for an arterial access. Limits of agreementare represented by the dotted line (mean of difference)and the two solid lines (±2S.D. of difference).

120 M. Prado et al.

Table 4 Limits of agreement (mean± 2S.D.) between the different methods used to calculate urea, nitrogengeneration rate (and BUN3est) and the reference value (see text)

Paired difference Limits ofagreement

95% Meanconfidenceinterval

95% 2S.D.confidenceinterval

P-value Standardskewness

StandardKurtosis

Gnest −G − 0.235± 0.420 − 0.235± 0.078 [0.335—0.565] 1.1× 10− 6 − 0.16 0.53(− 4.02± 7.18)% (− 4.02± 1.33)% [5.73—9.66]%

Gsp −G 0.484± 0.519 0.484± 0.097 [0.414—0.698] 4.18× 10− 11 2.75 2.38(8.27± 8.87)% (8.27± 1.66)% [7.08—11.93]%

BUN3est −BUN3 1.62± 2.62 1.62± 0.5 [2.1—3.5] 2× 10− 7 − 2.02 0.51(6.24± 10.10)% (6.24± 1.93)% [8.10—13.49]%

Percentages are referred to the mean urea nitrogen generation G= 5.85 mg/min (mean BUN3=25.95 mg/dl forlast row). Generation rate are in mg/min, and BUN in mg/dl. The 95% confidence interval for mean and 2 S.D. arealso presented to show the accuracy of the limits of agreement.

Finally, urea nitrogen generation rate estimatedby nspUKM when BUN3 is not known (Gnest), wasslightly undervalued (− 4.02± 7.18%, P< 10− 5).

5. Discussion

The study has been mainly focused on the develop-ment of a novel nspUKM, with the ability to computea better estimate of dKt/V than that obtained fromthe standard spUKM. We have tried to do the minorassumptions about the urea kinetic system. Accord-ing to that, and considering that the average ureanitrogen generation rate seems to be increased dur-ing intermittent therapy of uremia [51], our math-ematical analysis considered different urea gener-ations for the non-homogenous period, Gt

nh, and thehomogenous period, Gt

h. This issue makes the anal-ysis a bit more complex, but on the other hand, ex-tends the applicability of our results. Particularly,in cases where the difference Gt

nh −Gth could be sig-

nificantly positive, the urea removal and urea clear-ance estimates could be undervalued, as indicatedin (9) and (8). Despite its potential interest, theability of our approach to get conclusions relatedto the difference Gt

nh −Gth needs to be more inves-

tigated.A key issue in this work was the selection of

the most proper physiological input parameter tothe gspUKM and derived models, from which Kt/Vindices are analyzed and computed. Although thestandard spUKM and many current works that man-age spUKMs [54,55], use k as an input parameter,we have preferred to use the urea distribution vol-ume at the end of dialysis, VT , for this task. Thisselection simplifies the implementation of nspUKMas a model for hemodialyzer prescription, wherethe goal will be the computation of an adequatedialyzer clearance to deliver a dialysis dose given

by eKt/V, URR, or any other urea removal measure-ment as remarked in Section 1. Moreover, becauseof the uniqueness of the Kt/V solution, the conclu-sions that we have obtained are not restricted tothis choice, i.e. an overvaluation/undervaluation ofkd is equivalent to an undervaluation/overvaluationof VT (taking kd as input).

Indeed, the selection of the input parameter isnot a serious issue concerning the implementationof nspUKM to obtain the Kt/V, on account of thevery low sensitivity of this dimensionless index bothto k and VT variation. The low sensitivity of Kt/Vwas tested by numerical simulations and can alsobe proved by means of a dimensionless analysis ofthe gspUKM, although this item exceeds the scopeof this paper.

Reliability of nspUKM is strongly dependent on theimplementation of the areas condition AUC(T−TSP,T)=AUCv(0, T), which defines the TSP parameter.The clinical study presented has shown that the ar-eas condition may be accurately implemented bythree standard BUN samples: BUN1, BUN2 and BUN3,defined in Section 3.

Moreover, our results have confirmed that BUN3can be adequately estimated from the eKt/Vest ob-tained by the rate equation [19], by means of acomputational procedure that we have developedand presented in this work (Appendix E). Resultsshowed that BUN3est overvalued 6.24% the mea-sured BUN3 (Table 4). This issue agrees with theundervaluation of eKt/Vest to the standard eKt/V of1.14% (Table 3). Indeed, these estimates could beimproved taking the BUN2 sample 20 s after the end(in contrast to the 60 s used in this work), accordingto the scope of applicability of the rate equation foran arterial access [38]. This also could explain whythe distribution of differences between eKt/Vest

and eKt/V is somewhat deviated from the nor-mality as indicated by the standard skewness and

A novel mathematical method based on urea kinetic modeling for computing the dialysis dose 121

kurtosis parameters in Table 3 (absolute values ofthese parameters are greater than 2). The possibil-ity to estimate eKt/V avoiding a third BUN samplehas been the objective of many currents researchworks [29,30,34,56,57]. This fact provides us otherpotential lines to improve BUN3est, nevertheless weselected the rate equation due to many clinicalworks that have demonstrated its high accuracy.

The scope of applicability of our model has beenverified for a range of estimated dKt/V between1.1 and 2, and dialysis durations between 3 and 4.5h, on stable ESRD patients. Nevertheless, we thinkthat the scope can be easily extended, e.g. to con-sider dialysis durations far away from the afore-mentioned interval. This is founded on the reach-able accuracy of the areas condition. Several is-sues concerning this issuemay be appointed. Firstly,Burgelman et al. [58] showed that the accuracyof the identification process of a double pool ki-netic model does not improve with more than sixBUN samples. Furthermore, the integral operator∫ ·dt greatly reduces the accuracy requirements forcv(t), on account of its behavior as low pass filter.This agrees with the reliability of the rate equation[19], which estimates eKt/V from spKt/V using onlythe rate constant k/V, showing that the urea re-bound is mainly described by first order dynamics.

The nspUKMwas implemented, solving iterativelythe algebraic system that results from the analyticalintegration of an ODE system, the same way as thestandard spUKM. The TSP parameter was obtainedfrom an explicit formula, keeping the simplicity ofthe computational procedure. However, we cannotuse bedside equations as the second generation for-mula from Daugirdas [11], to obtain the nKt/V in-dex. For example, we have verified that the sub-stitution of T by Tsp in this latter formula providesa bad estimate of nKt/V. Despite this fact, we arepreparing a modification of the second generationformula that will be able to be applied as a bed-side formula to calculate nKt/V, keeping the bene-fits that bedside equations provide to the solutionof spUKMs in clinical environments.

A particularly interesting result is the slight un-dervaluation of 2pKt/V by spKt/V. Moreover, weverified that spKt/V was closed to 2pKt/V for amean Kt/V value near 1.25 (see Fig. 9), and theundervaluation grows with the mean Kt/V. Theseresults agree with [59], where authors analyticallyfound that for Kt/V values about 1.3, the single-pooland double-pool dialyzer clearance estimates co-incide, while the difference makes more negativewhen Kt/V increases. Moreover, spKt/V overvalueseKt/V in about a 12% (see Table 3) what agrees withother works [38,42,23]. Statistical distribution ofthe difference between spKt/V− 2pKt/V deviates

from normality, as shown in Table 3, and as a conse-quence the 95% confidence interval for its S.D. maybe inaccurate.

The assumption that the equilibrium of urea isreached at tf =T+ tequ, where tequ = 30 min, needsmore justification. Many clinical works have veri-fied that the urea rebound finishes about 30 minafter the end of dialysis, nevertheless a slight dis-persion may occur between different patients andHD sessions. With the aim to obtain the changeof 2pKt/V when tf is altered, the 2pUKM wasagain fitted, modifying the equilibrium conditionsuch that this was reached at t′f = tf − 10 min.The percentage of variation of 2pKt/V, calculatedas 100 ·(2pKt/V(t′f)− 2pKt/V(tf))/2pdKt/V(tf) was− 0.5± 0.6%. Therefore, the sensitivity of 2pKt/Vto tf was − 0.05± 0.6%/min. These results confirmthe validity of the aforementioned assumption.

The ratio between accessible and non-accessiblecompartments (V1/V2) for this second 2pUKM im-plementation was 0.38± 0.16, what differs from0.6± 0.23 of the first implementation. This factconfirms the expected inaccuracy of 2pUKMs to es-timate physiological volume fractions, due to thefact that regional blood circulation mechanism isnot managed by them [20].

Finally we must remark that when AR is high,nKt/V will be actually related to an effective dia-lyzer clearance that embraces the dialyzer togetherwith the AR efficiency. This fact shows another prac-tical interest of nspUKM in HD quantification. Thelatter is currently being developed by our group.

6. Conclusions

Wehave presented a novel urea kineticmodel whichjoins together the simplicity of the single-pool mod-els with the accuracy of double pool models, whenthe target is not the BUN evolution but the com-putation of the dKt/V, the urea generation and theurea removal during a dialysis session.

Analytical and experimental results have demon-strated that the standard spUKM overvalues ureageneration and urea removal. This overvaluationis proportional to the urea rebound. Deviationbetween spKt/V and dKt/V is a consequence ofthe unbalance between urea inbound and urearebound, which has been quantified in (B.2) andconfirmed experimentally. Under the scope of theclinical study, the urea inbound dominates whendKt/V is greater than a value about 1.3.

Urea generation and urea removal estimates arecorrected when spUKM is used for the calculationof eKt/V. Despite this index is a very good measureof the patient dialysis dose, it strongly undervalues

122 M. Prado et al.

dKt/V and with high dispersion, because of the ureainbound (B.5), and therefore, it cannot be used topredict or assess the dialyzer clearance.

We have validated the high accuracy of nspUKM toestimate dKt/V. This model overcomes limitationsimposed by the urea disequilibrium induced by HDto standard single-pool models. Two implementa-tions have been described and tested. The first ofthem used three standard BUN samples, while thesecond only required two BUN samples. This modelcan not be used to predict urea dynamics, but thislimitation can be accepted within clinical environ-ments. Currently, the knowledge of urea dynam-ics is only important for research purposes. We arenow preparing a bedside formula for the physicians,which will simplify furthermore the calculation ofa dKt/V estimate.

7. Glossary

The major symbols are presented below. Dimen-sions are indicated by L for length, t for time and Mfor mass. The superscript † remarks target or knownvariables in this study, i.e. variables that are notcalculated by any model.

VT postdialysis urea distribution volume(L3)

UF weight loss during HD session (L3)δV(τ) weight gain during τ units of time (L3)fv1 fraction of accessible compartment

volumeV1, V2 accessible and non-accessible body

volumes (L3)Q UF ultrafiltration flow rate (L3 t− 1)†

Q b blood flow rate (L3 t− 1)†

T dialysis session duration (t)†

TSP gspUKM dialysis session duration (t)tequ equilibrium interval duration (t)†

τ delay for blood drawing since the endof dialysis (t)†

θi interdialysis duration following dialysisi (t)†

p patient assessment temporal interval(t)†

tl predialysis instant from dialysis l (t)†

G average urea nitrogen generation (Mt− 1)

Gth true average urea nitrogen generation

at [T+ tequ, p] (M t− 1)†

Gtnh true average urea nitrogen generation

at [0, T+ tequ] (M t− 1)†

Gt true average urea nitrogen generation(M t− 1)†

m urea removal rate (M t− 1)Pu net urea nitrogen loss (M)Ptu true net urea nitrogen loss (M)†

kd dialyzer clearance (L3 t− 1)†

k model clearance (L3 t− 1)Kc compartmental urea nitrogen

clearance (L3 t− 1)c urea nitrogen concentration (M L− 3)cv vascular access urea nitrogen

concentration (M L− 3)†

ct(x, t) patient urea nitrogen concentration atposition x (M L− 3)†

c1, c2 accessible and non-accessible ureanitrogen concentration in 2pUKM (ML− 3)

BUN BUN sample (M L− 3)†

Dimensionless numbersKt/V fractional urea clearancespKt/V single-pool Kt/VnKt/V normalized Kt/VeKt/V equilibrated Kt/V2pKt/V two-pool Kt/VdKt/V dialyzer Kt/V †

AcronymsspUKM single-pool Urea Kinetic ModelespUKM equilibrated spUKMgspUKM generalized spUKMnspUKM normalized spUKMESRD end-stage renal diseaseHD hemodialysisMW molecular weightAR access recirculation

Appendix A. Urea inbound

We wish to obtain a mathematical relationship be-tween c(t) (single pool system) and cv(t) (patient)in an HD session, when the urea distribution volumeV(t), the urea nitrogen generation rate G and thedialyzer clearance k are the same for both systems.

If the urea is homogeneously distributed at t= 0into the patient and c(0)= cv(0), the following as-sertion is true:∫ t′

0cv (t)dt︸ ︷︷ ︸

AUCv (0, t′)

≤∫ t′

0c(t)dt︸ ︷︷ ︸

AUC(0, t′)

, (A.1)

where t′ is any time t≥0. We call urea inbound thedifference AUC(0, t′)−AUCv(0, t′). Equation (A.1)may be proved considering that there are no ac-

A novel mathematical method based on urea kinetic modeling for computing the dialysis dose 123

tive transport mechanisms for the urea, and so theurea transport is governed by the Fick diffusion law.Therefore, the urea nitrogen concentration at anyplace x into the patient urea distribution volume,ct(t, x), must be greater or equal than cv(t). Consid-ering this fact, the equation may be easily provedif we suppose the converse, i.e. that there existsany t′ such that AUCv(0, t′)>AUC(0, t′). Under thiscondition, if c(t) and cv(t) are continuous functions,then function F(t)=AUCv(0, t)−AUC(0, t) will alsobe continuous. Therefore, considering that F(0)= 0,then there must be a tχ where F(tχ)= 0 and F(t)> 0for all t∈(tχ, t′].

Since F(tχ)=AUCv(0, tχ)−AUC(0, tχ)= 0, andtaking into account that k is equal in both systems,then the total urea removal from t= 0 to t= tχ willbe equal for both systems. As F(t)> 0 for all t∈(tχ,t′] then the total urea removal will be greaterin patient than in single-pool system, and so theremaining total urea will be greater in the last sys-tem. This can be written by the right inequality ofthe following equation:∫V(t)

cv (t)d� ≤∫V(t)

ct(t, x)d� <∫V(t)

c(t)d�

where we have used volumetric integrals, and sod� is a differential of volume. The left inequality isa consequence of the Fick law. Taking into accountthat c(t) and cv(t) do not depend on x, we deducethat cv(t)<c(t) for all t∈(tχ, t′], and hence AUCv(0,t′)<AUC(0, t′). The latter is contradictory with theaforemention supposition, what proves (A.1).

Appendix B. Analytical study of spKt/Vand eKt/V indices

The models which define the standard spKt/V andeKt/V indices can be obtained assigning proper val-ues to the parameters τ and TSP of the gspUKM.These models will be analyzed to know the devia-tions of urea clearance k, urea nitrogen generationrate, G, and urea removal estimates from targetvalues.

B.1 spKt/VThe spKt/V is obtained from the gspUKM, by as-

signing to the delay τ a value between 20 s and 1or 2 min, and TSP =T. Under this delay range, theAR component of urea rebound is always removed[38]. Fig. 12 illustrates the evolution of c(t) underthese conditions.

In this model, the difference (6) is positive be-cause of the urea rebound induced by the HD ses-sion. Therefore, (7) proves that the urea nitrogengeneration rate G estimated by the spUKM over-

Fig. 12 Temporal evolution of cv (t) (dotted line) andc(t) (solid line) for the standard spUKM, indicating the bcassignments. It has been considered τ= 0 to clarify theillustration.

values the true average urea nitrogen generationrate Gt

h. Although Gtnh seems to be increased dur-

ing intermittent therapy of uremia [51], we canconsider Gt

h ≈Gt because θ T. This overvaluationagrees with previous works, e.g. the comparativestudy from Capello et al. [54].

Equation 8 proves that this spUKM overestimatesurea removal, which also agrees with previousworks [38,46].

Finally, the accuracy of k as an estimate of kd,is deduced from (8). Neglecting low order termsin (8), i.e. assigning Gt =Gt

h =Gtnh, p− (T+ tequ)≈p,

δV�VT and c(T+ tequ)≈ c(T)≈ cv(T+ τ), (8) can berewritten as:∫ T

0(kc(t)− kdcv (t))dt=VT (cv (T + tequ)−cv (T + τ))︸ ︷︷ ︸

Pu−Ptu

⇒ cv (T + tequ)− cv (T + τ)

= kdTVT

1T

∫ T

o

(kkd

c − cv

)dt (B.1)

From (B.1) follows that spKt/V is equal to dKt/V,i.e. k= kd, if and only if:

cv (T + tequ)− cv (T + τ)︸ ︷︷ ︸rebound

= kdTVT

(c − cv )︸ ︷︷ ︸inbound

, (B.2)

where c − cv is the time average of the differencec(t)− cv(t) during the HD interval (urea inbound).The left member of (B.2) is the urea rebound, whichas commented previously does not include AR, dueto the selection of τ≥20 s.

From (B.1), when the urea rebound is greaterthan dKt/V times the urea inbound, k overvalues kd.The reciprocal is also true. Therefore, the overval-uation or undervaluation of kd is governed by twoopposite terms: the urea rebound and the urea in-bound.

B.2 eKt/VIndex eKt/V is obtained from the gspUKM, assign-

ing TSP =T and τ= tequ. Fig. 13 shows the evolution

124 M. Prado et al.

Fig. 13 Temporal evolution of cv (t) (dotted line) andc(t) (solid line) for the espUKM, indicating the bc assign-ments.

of c(t) under these conditions. We refer to thismodel (including the bc) as espUKM.

The difference Pu − Ptu is now null and thus (7) and(8) may be written as follows.

∫ T

0(kc(t)− kdcv (t))dt= (T + tequ)(Gt

h − Gtnh)

(B.3a)

G=Gth (B.3b)

Equation (B.3a) proves the urea removal is accu-rately estimated by espUKM, although its precisioncan be perturbed by the difference Gt

h −Gtnh. Equa-

tion (B.3b) proves that G is an accurate estimate ofGt

h ≈Gt.With regard to the k estimate, and using (A.1) re-

sult with G and kd as parameters for both systems,cv(0) as initial condition for the urea nitrogen con-centration and V(t) as urea distribution volume, thefollowing equation results:

0 <∫ T

0(kdc(t)− kdcv (t))dt (B.4)

where c(t) is the single-pool urea nitrogen concen-tration resulting under these conditions. Consid-ering Gt

h −Gtnh≤0, according to [51] in (B.3a) and

placing the resulting equation together with (B.4),the following inequalities result:

∫ T

0(kc(t)− kdcv (t))dt ≤ 0 ≤

∫ T

0(kdc(t)− kdcv (t))dt

⇒∫ T

0kc(t)dt ≤

∫ T

0kdc(t)dt ⇒ k ≤ kd (B.5)

The last step in (B.5) is due to the increase ofurea removal when k grows, keeping the remain-ing conditions. This proves that eKt/V undervaluesdKt/V, and this deviation is mainly caused by theurea inbound, which is quantified by the differenceAUC(0, T)−AUCv(0, T).

Appendix C. Implementation of theareas condition

The areas condition AUC(T−TSP, T)=AUCv(0, T) pro-vides an integral equation to solve TSP. The objec-tive of this section is the development of a goodenough approximation for cv, called cv est, startingfrom BUN1, BUN2 and BUN3 samples, which gives anaccurate and explicit formula for TSP as a solutionof the areas condition. Taking into account that adetailed description of this estimate would extendtoo much this paper, whose objective at this pointis the development of an experimentally validatedformula for TSP, the present section is limited topresent and briefly justify the several phenomeno-logical considerations used. Urea nitrogen concen-tration, cv est, is defined as follows:

cv est(t)=

BUN1e−t/τ1 t ∈ [0, tequ]BUN3

e−a·T/τsp e−at/τsp +BUN2 − BUN3

t ∈ [tequ, T]

(C.1)

As indicated by (C.1), two different regions in theurea dynamics during an HD session have been con-sidered. The first matches a fast depletion, wherethe difference between non-accessible urea con-centration and cv est grows, while this difference de-creases at the second region. The first region is keptfor a time t=tequ according to the characteristictime of urea rebound. Both regions are governed byfirst order dynamics, although they have differentrates, τsp/a and τ1, given by τsp=T/ln(BUN1/BUN3)and τ1=tequ/ln(BUN1/cv est(tequ)), and being a a scalefactor function below 1. All these equations havebeen developed to fulfil with the bc cv est(0)=BUN1,cv est(T)=BUN2, together with the fast depletion atthe beginning of HD. The latter is controlled by a,which has been fitted by the following regressionmodel:

a(R, Re)=β1 ln(Re)+β2 ln(R)+β3 (C.2)

where Re =BUN3/BUN1 and R=BUN2/BUN1. En-countered predictor variables showed that a fol-lows approximately equation 1 − ln(BUN3/BUN2),i.e. it is governed by urea rebound, as expected.

Using a single-exponential approximation for theurea nitrogen concentration of the nspUKM, cest, ac-cording to the bc defined in Section 2.2, and sub-stituting cv est and cest into the areas condition, weobtain the desired equation:

TSP =A(R, Re) · T +B(R, Re) · tequ, (C.3)

where A and B are as follows:

A(R, Re)=(Re − R)ln(Re)+ ψ

a1 − Re

(C.4a)

A novel mathematical method based on urea kinetic modeling for computing the dialysis dose 125

B(R, Re)= ln(Re)1 − Re

(R − Re − ψ − (1 − R)

ln(R+ψ) ,)

(C.4b)

being ψ=Re(Ra(tequ/T− 1)e − 1).

Linear regression for a(R, Re) (C.2) was firstlycomputed for a validation group of 15 patients ran-domly selected from the whole study group. Thismodel was applied afterwards to a cross-validationgroup formed by the remaining 15 patients, com-paring the RMSE of the validation group with thatof the cross-validation one, with the aim to mea-sure the performance of the model. The finalprediction equation was developed as a linear re-gression for the total of patients of this study.Statistical parameters as R2, hypothesis testsand influence analyses have been presented inSection 4.

Appendix D. Two-pool urea kineticmodel

This study uses a simplified 2pUKM verified byCanaud et al. [23]. In this model V1 and V2 repre-sent the accessible and non-accessible urea distri-bution volumes, G is the urea nitrogen generationrate, Kc is the intercompartmental urea mass trans-fer coefficient, and kd is the effective dialyzer ureaclearance. The model is defined by the followingequations:

X1(t)= − kdc1(t)+Kc(c2(t)− c1(t))+G (D.1a)

X2(t)= − Kc(c2(t)− c1(t)), (D.1b)

where X1(t) and X2(t) are compartment ureamasses. They are related to accessible urea con-centration c1(t)=X1(t)/V1 and non-accessibleurea concentration c2(t)=X2(t)/V2. Urea distri-bution volumes are defined as V1(t)= fv1V(t) andV2(t)=V(t)−V1(t), being V(t) the total urea dis-tribution volume and fv1 the fraction of accessiblevolume. This is defined as V(t)=VT +UF−Q UF · t,during the HD session.

The following variant of the previous two-pool model (D.1) was implemented to test thevery low influence of the water transport in2pKt/V:

X1(t)= − kdc1(t)+Kc(c2(t)− c1(t))+ ν+G (D.2a)

X2(t)= − Kc(c2(t)− c1(t))− ν, (D.2b)

being ν= (1− fv1)Q UFc2.

Appendix E. Solution of the nspUKM

The analytical solution of (2) is as follows:

c(t, �) = G

(k+ β)+(c(0)− G

(k+ β)

)

·(V(0)+ βt

V(0)

)−(k+ β)/β(E.1)

where�= (c(0), k, β, G, V(0) is a parameter vector.The urea nitrogen generation rateG, and the urea

nitrogen clearance k is solved from (E.1) togetherwith (3), applying the boundary and cc (4) and (5).We assigned cv(p)= cv(0), according to the two BUNmethod, obtaining the following equation system:

c(Tsp, C01, k, −QUF

TTSP

, GTTSP

, VT +QUFT)

=CT1

c(θ1, CT1, 0, QUF

Tθ1, G, VT

)=BUN1

c(Tsp, BUN1, k, −QUF

TTSP

, GTTSP

, VT +QUFT)

= CT2(G)

c(θ2, CT2(G), 0, QUF

Tθ2, G, VT

)=C03

c(Tsp, C03, k, −QUF

TTSP

, GTTSP

, VT +QUFT)

=CT3

c(θ3, CT3, 0, QUF

Tθ3, G, VT

)=C01

CT2(G)=BUN3VT +QUF

Tθ2tequ

VT− Gtequ

VT(E.2)

where CTi and C0i are respectively the single-poolpostdialysis and predialysis urea nitrogen concen-trations of a session number i, being i= 1, 2, 3. Thelast equation of (E.2) represents the second bc in(4).

Solving iteratively (E.2), parameters C01, CT1, C03and CT3, together with G and k, are obtained. Singlepool dialysis duration, TSP, is solved previously from(10), (C.3) and (C.4).

Regarding the obtention of nKt/Vest, which isthe nKt/V estimated when BUN3 is not known butcalculated from BUN2 and BUN1, we have devel-oped a computational procedure arranged into fourstages. First, an accurate estimation of spKt/V isobtained from the Daugirdas second generationformula (G.1). The eKt/Vest is obtained afterwards,by means of (G.2). Since VT is known from theChertow formula, the whole body clearance ek canbe solved from eKt/Vest. At third stage, the system(E.2) with TSP =T, ek as an input parameter, and

126 M. Prado et al.

CT2 as an output parameter, is solved. BUN3est iscalculated from CT2 by means of the last equationin (E.2). The nspUKM is run at the last stage, us-ing BUN3est to solve TSP from (C.3) and (C.4) andsubstituting CT2(G) in (E.2) by the previous solvedparameter CT2.

Appendix F. Solution of the spUKM

The eKt/V index can be solved from (E.2), in a simi-lar manner than nKt/V, taking care of assigning pre-viously TSP =T, as is described in Appendix B.2.

The same equation system (E.2) is used to ob-tain spKt/V, taking care of switching BUN3 by BUN2and tequ by the proper delay τ, as is described inB.1. Considering that τ� tequ for a standard post-dialysis BUN sample, the last equation in (E.2) canbe substituted by CT2(G)=BUN2. Indeed, the stan-dard spKt/V was defined with this latter bc [6], andtherefore, this was applied in the clinical study pre-sented.

Appendix G. Other formulae

Taking into account the previous definition of R in(C.2), the second generation formula of Daugirdas[11] can be written as follows:

spKt/V= − ln(R − 0.008

T60

)+ (4 − 3.5R)

UFW

(G.1)

where W is the dry-weight (kg), T is the HD period(minutes) and UF is the total weight loss during HD(l). This formula provides a high accurate estimateof the spKt/V calculated by the standard spUKM.

The rate equation of Daugirdas and Schneditz [19]is given as follows:

eKt/V= spKt/V(1 − 0.6

T

)+ 0.03 (G.2a)

eKt/V= spKt/V(1 − 0.47

T

)+ 0.02 (G.2b)

where T is in hours. The first equation in (G.2) isused for an arterial vascular access, while the sec-ond is valid for a venous access. The accuracy of(G.2) has been verified by several authors, guaran-tying that the postdialysis sample delay τ is about20 s for an arterial access and between 20 s and 2min for a venous access [38].

Total body water TBW can be used as a good esti-mation of VT . It is computed in this paper from theChertow et al. regression [49]:

TBW = − 0.07493713E − 1.01767992S

+ 0.12703384H − 0.04012056W

+ 0.57894981D − 0.00067247W2

− 0.03486146E · S+ 0.11262857S · W+ 0.00104135E · W + 0.00186104H · W

(G.3)

whereWwas defined in (G.1),H is the height (cm), Eis the age (years), S= 1 for males and 0 for females,and D= 1 for diabetic patients and 0 otherwise.

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