A two-pool kinetic model predicts phosphate concentrations during and shortly following a conventional (three times weekly) hemodialysis session

Abstract

Background

Previous studies have suggested that a conventional two-pool model cannot be used to predict intradialysis and early postdialysis phosphorus concentrations.

Methods

A conventional two-pool urea model was modified by increasing the distal compartment volume from two-thirds to three times the total body water and by the use of a dynamically variable intercompartmental phosphorus clearance during dialysis. The phosphate solver model parameters were derived from an examination of the results in the literature, and fine-tuned using a training set (F4) of 415 Hemodialysis (HEMO) Study patients studied during a dialysis session where phosphorus was measured at 4 months of follow-up. Validation was done in a group of 380 different HEMO Study patients plus 9 from the original F4 group, who were evaluated at 36 months of follow-up.

Results

The model predicted measured median early (1 h) intradialysis, end-dialysis and 30-min postdialysis serum phosphorus levels in the test and validation datasets with little apparent bias, including the highest and lowest deciles of predialysis serum phosphorus. The model tended to underestimate slightly intradialysis serum phosphorus when predialysis serum phosphorus was <3.0 mg/dL (0.97 mmol/L). There was a large scatter and standard deviation among patients, and whether aberrant values represent a patient-specific phenomenon is unclear.

Conclusions

A modified two-pool model using a slightly expanded distal compartment and a dynamically varying intercompartmental clearance, depending on the intradialysis phosphorus concentration, can be used to predict serum phosphorus level during and shortly after dialysis, in patients following a conventional three times per week dialysis prescription.

INTRODUCTION

The success of urea kinetic modeling inspired similar efforts to describe phosphorus kinetics. A number of such models have been proposed. Early reports suggested that a standard model was not applicable to phosphorus removal and could not be used to predict postdialysis phosphorus rebound [1, 2]. Spalding et al. [3] found that a urea-type two-pool model where the proximal and distal compartments were one-third and two-thirds of the total body water, respectively, and where intercompartmental clearance was approximately 350 mL/min, was only moderately useful and tended to underestimate the postdialysis serum phosphorus value in those patients with a low predialysis serum phosphorus. They attempted to solve the problem by adding two additional sources of phosphorus that would become activated once critically low intradialysis levels of serum phosphorus had been reached. Gotch et al. [4] described a model of phosphorus removal based on patient-specific measures of intradialysis phosphorus transport. Agar et al. [5] and Leypoldt et al. [6] attempted to solve the problem of underestimation of postdialysis serum phosphorus by the use of a two compartment model with an infinitely large distal compartment and a fixed, patient-specific intercompartmental clearance. More recently, Debowska et al. [7] and Polesczuk et al. [8] have attempted to define weekly measures of phosphorus removal and to address the issue of the plateau effect of intradialysis serum phosphorus during very long dialysis sessions.

The goal here was to create a model that could be used to generate a complete weekly serum phosphorus profile that could be used not only to predict intradialysis and postdialysis serum concentrations, and how predialysis serum concentrations might be altered by changes in dialysis therapy for any given level of estimated phosphorus absorption. We adapted a two-pool diffusion-based model to help predict intradialysis and early postdialysis serum phosphorus values.

MATERIALS AND METHODS

Phosphorus kinetic model

In the phosphorus model, a conventional two-pool urea model [9, 10] is used, but with several modifications. Like the urea model, the phosphorus model is diffusion based, but involves a distal storage pool that is somewhat larger than the total body water. In simulations, the ideal size for the distal phosphorus pool was examined. When the distal pool is approximately equal to the total body water, or Vurea, the phosphorus reduction ratio exceeds that which is found clinically, and when predialysis serum phosphorus levels are low, the postdialysis serum phosphorus value is substantially lower than observed. There needs to be more exchangeable phosphorus in the pool than can be accounted for by the intracellular space. On the other hand, expanding the distal storage pool beyond four times Vurea results in loss of the normal variation in predialysis serum phosphorus with asymmetric schedules. In a three times per week schedule, the degree of dependence of predialysis serum phosphorus on the preceding interdialytic interval is somewhat unclear. Fujimoto et al. [11] and Shimazaki et al. [12] both found a 0.3 mg/dL (0.097 mmol/L) higher predialysis serum phosphorus at the beginning of the week after a 3-day interval than prior to the midweek dialysis session. In the FHN Daily Trial, where the analysis was based on repeated measures of predialysis serum phosphorus, the average difference in predialysis serum phosphorus per 2-day interval was 0.49 mg/dL (0.16 mmol/L), and there was evidence that the difference between a 1-day and 2-day interval was substantially greater than between a 2-day and a 3-day interval [13]. On the other hand, Debowska et al. [7] studied 25 patients during each of three dialysis sessions during the week and found no difference between first-of-the-week and midweek dialysis serum phopshorus levels. In the Hemodialysis (HEMO) Study, predialysis variation in serum phosphorus across the week is available only in cross-sectional data. In the HEMO dataset, there was no significant difference between first-of-the-week and midweek predialysis serum phosphorus values.

The relationship between the size of the distal phosphorus storage volume and the differences between predialysis serum phosphorus values for an asymmetric three times per week schedule are shown in the Supplementary data, Appendix (see Supplementary data Figure). On the basis of an analysis of rebound in our model as well as weekly variation in predialysis serum phosphorus in asymmetric schedules, we decided to set the size of the distal phosphorus storage pool at three times the total body water value (or at 4.5 times the modeled intracellular water space).

The remaining two important adjustments in the two-pool model are how to size the volume of the proximal compartment from which phosphorus is first removed and how to set the value for the intercompartmental clearance (Kc). The size of the proximal compartment impacts the degree to which the intradialysis serum phosphorus value decreases early during dialysis and also on the extent of the early postdialysis urea rebound. The value for Kc affects the degree of phosphorus reduction during a dialysis session. A higher value for Kc will result in a higher value for postdialysis serum phosphorus, as it allows more ‘refilling’ of phosphorus from the distal pool during dialysis. The optimal value for Kc will change slightly, based on the ratio of the distal P storage pool to the urea volume. One additional factor to consider is the dialyzer phosphorus clearance. Higher dialyzer clearances will reduce intradialysis serum phosphorus levels at all time points.

In preliminary data, we confirmed the data of Spalding et al. [3], that use of a fixed value of Kc resulted in underestimation of postdialysis serum phosphorus. To overcome this limitation, in our model we adjusted the Kc during dialysis in a dynamic fashion, leaving it at a fixed value when serum phosphorus was above a set point, which we estimated to be somewhere around 3.0 mg/dL (0.97 mmol/L), and then increased the Kc value steeply as the intradialysis serum phosphorus fell below the set point value.

On the basis of review of data in the literature and considerations above regarding the size of the distal compartment, we initially set the volume of the proximal compartment at 0.33 and the Kc value at 100 mL/min, with a set point of ∼3.5 mg/dL. On the basis of the literature, we estimated dialyzer phosphorus clearance to be based on a K₀A (mass transfer area coefficient) for phosphorus that was 0.4 times the K₀A for urea [5, 6]. It was assumed that no phosphorus was removed from red blood cells during passage through the dialyzer [14]. The plasma water and blood water coefficients used in the Michaels equation [see 9] were 0.93 and 0.86, respectively, and the ratio of to whole blood plasma flow rates was 0.67.

In fine-tuning the values for proximal compartment V and Kc, we used two sets of dialysis data available from patients in the HEMO Study, one at 4 months of follow-up (F4) after randomization, which was taken to be a ‘training set’, and the other at 36 months of follow-up (F36) after randomization, which was taken to be a ‘validation set’. At these two time points, in addition to the usual monthly predialysis and postdialysis blood samples, one additional blood sample was obtained 1 h into dialysis and a second additional delayed postdialysis blood sample was taken 30 min after dialysis.

Initially in the HEMO Study, the peridialysis blood samples from F4 and F36 were sent for urea and creatinine analysis, but not for phosphorus. After the HEMO Study was well under way, the decision was made to also measure phosphorus in these peridialysis blood samples. A consequence of this decision was that many patients enrolled in the HEMO Study during the initial period of the recruitment had no phosphorus values during their F4 study, thus these early recruited patients had phosphorus data only at 36 months of follow-up. Similarly, a large group of HEMO patients who enrolled during the latter part of the study period had phosphorus data from their extended dialysis modeling session at 4 months of follow-up, but overall follow-up in the study was terminated before these later-enrolled patients could reach their 36-month follow-up point. For this reason, in the two-patient datasets analyzed at months F4 and F36, there were only nine patients common to both time groups with analyzable phosphorus data, and thus the two groups were treated as independent samples.

RESULTS

Patient and treatment details

The demographic and treatment characteristics of the F4 and F36 HEMO patients, divided by sex, are illustrated in Table 1. The differences between men and women are apparent. The F36 patient group had slightly higher values for blood flow rate and dialyzer clearances; most of this was probably due to the fact that the F36 group had a higher percentage of males than the F4 group (46 versus 40%, respectively). Note that use of an estimated clearance ratio of 0.4 for phosphorus:urea K₀A resulted in an average phosphorus:urea diffusive dialyzer clearance ratio close to 0.58.

Table 1.

Demographic and dialysis treatment-related variables for HEMO Study patients at 4 months and 36 months of follow-up (F4 and F36, respectively) that included phosphorus measurements

	F4 group				F36 group				P-value, F4 versus F36
	Women	Men	P-value, women versus men	Both genders	Women	Men	P-value, women versus men	Both genders
No. of cases	247	168		415	210	179		389
Postdialysis weight (kg)	68.1 (16)	73.6 (14)	<0.001	70.3 (15)	65.7 (14)	72.6 (16)	<0.001	68.9 (15)	0.21
Age (years)	59.3 (14)	58.0 (15)	0.371	58.8 (15)	60.9 (13)	57.1 (15)	<0.001	59.1 (14)	0.70
URR (%)	71.6 (6.4)	70.0 (6.5)	0.013	70.1 (6.5)	71.5 (6.8)	70.3 (6.4)	0.075	71.0 (6.6)	0.95
Kdif-urea (mL/min)	224 (33)	238 (32)	<0.001	229 (33)	229 (29)	243 (27)	<0.001	235 (28)	0.0056
PRR (%)	55.9 (9.42)	49.7 (12.8)	<0.001	53.2 (11.3)	54.7 (10.4)	52.6 (12.4)	<0.001	53.8 (11.3)	0.336
Kdif-phos (mL/min)	131 (18)	137 (19)	<0.001	134 (18)	133 (14)	140 (16)	<0.001	136 (15)	0.035
Kdif-phos /Kdif-urea	0.59 (0.031)	0.58 (0.023)	<0.001	0.58 (0.024)	0.58 (0.026)	0.58 (0.022)	<0.001	0.58 (0.025)	0.014
Qba (mL/min)	323 (66)	353 (55)	<0.001	335 (64)	337 (61)	362 (50)	<0.001	349 (58)	0.0021
Qd (mL/min)	679 (131)	676 (132)	0.80	678 (131)	680 (125)	703 (119)	0.10	691 (123)	0.136
Td (min)	199 (25)	218 (27)	<0.001	207 (28)	197 (27)	219 (30)	<0.001	207 (31)	0.89
Modeled V (L)	29.0 (5.9)	35.6 (7.6)	<0.001	31.7 (7.4)	29.7 (7.8)	36.0 (7.4)	<0.001	32.5 (8.3)	0.094
Modeled G (mg/min)	4.41 (1.5)	5.46 (1.8)	<0.001	4.83 (1.7)	4.05 (1.40)	5.10 (1.75)	<0.001	4.54 (1.66)	0.013
Modeled phosphorus removed (mg)	970 (302)	1105 (395)	<0.001	1024 (345)	961 (346)	1150 (396)	<0.001	1050 (382)	0.309

Kdif-urea, dialyzer diffusive urea clearance; Kdif-phos, dialyzer diffusive phosphorus clearance. URR, urea reduction ratio; PRR, phosphorus reduction ratio.

^aThe value for Qb is not the nominal value, but rather the corrected flow rate, adjusted for blood line oval deformation due to high prepump pressure. The correction is according to a correction equation used in the HEMO Study.

Optimization of Kc and Vproximal

The adjustments to Kc and Vproximal in the F4 ‘training set’ patients are presented in Table 2. These adjustments were made in an iterative fashion and were based on optimizing the percent error of the postdialysis and 1-h intradialysis serum phosphorus values. We found early on that Kc should be adjusted to body size, because without this adjustment, the percent error in postdialysis serum phosphorus was significantly related to urea distribution volume (V; V in this article means postdialysis two-pool urea distribution volume). We reflected this body size association by adjusting the value for Kc, multiplying it by V/35. With a V of 35 liters, an optimal value for Kc was found to be 86 mL/min in men, but Kc needed to be reduced substantially in women. Multiplying the V-normalized Kc by 0.83 in women removed this bias. We adjusted the ‘set point’ for when Kc should begin to increase by examining the percent error in postdialysis serum phosphorus according to the deciles of predialysis serum phosphorus. Our initial guess for a set point in the 3.5 mg/dL (1.13 mmol/L) range was modified: results were better when the set point was reduced slightly to 3.0 mg/dL (0.97 mmol/L). The magnitude of the increase in Kc below the set point was adjusted by examining the percent error in postdialysis serum phosphorus in the three lowest deciles of predialysis serum phosphorus. The relationship between Kc and the extracellular phosphorus concentration during dialysis that was determined empirically in this fashion is shown in Figure 1.

Table 2.

Modeling parameters based on HEMO Study patients at 4 months of follow-up (n = 415)

Number of pools	Size of the distal pool	Size of proximal pool		Kc (mL/min/35L V) dynamically adjusted during the dialysis treatment
Both genders	Both genders	Men	Women	Men		Women
				P < 3.0 mg/dL (0.97 mmol/L)	P ≥ 3.0 mg/dL (0.97 mmol/L)	Any range of P
2	3.0 V	0.26 V	0.32 V	686‒200	86	As for men, but multiplied by 0.83

V = postdialysis two-pool urea distribution volume; P = intradialysis serum phosphorus.

FIGURE 1.

In the proposed model, the value of Kc changes dynamically, every second, based on the modeled level of extracellular phosphorus (P) as controlled by the template above. As long as the extracellular intradialysis phosphorus level is ≥3.0 mg/dL (0.97 mmol/L), the normalized Kc value is set to 86 mL/min/35 L total body water (for males). The Kc value is adjusted for body size by multiplying by the ratio of postdialysis urea volume). When the serum phosphorus levels drops below 3.0 mg/dL (0.97 mmol/L), the Kc value is increased according to the empirically developed equation shown, such that the Kc is ∼300 mL/min when the momentary extracellular phosphorus is 2.0 mg/dL (0.65 mmol/L). In women, the Kc value (across the whole range of extracellular phosphorus) is reduced, multiplying the value for males by the empirically determined factor of 0.83.

With regard to the best size of the proximal phosphorus pool, iterative analysis of percent error of the 1-h intradialysis sample suggested that the optimal values for Vphosphorus-proximal were 0.32 × Vurea in women and 0.26 × Vurea in men.

Analysis of error in the predicted postdialysis serum phosphorus

Figure 2 shows the percent error in the F4 ‘training’ sample of patients plotted against several potential confounders, including postdialysis Vurea, age, predialysis serum phosphorus and dialysis session length. As shown in the figure, there was nothing approaching a trend to suggest that further correction for any of these factors would improve prediction accuracy. In the HEMO Study, there were also a substantial number of patients with residual kidney urea clearance up to 2.0 mL/min. Again, the percent error in prediction of postdialysis phosphorus was not related to the relatively modest amounts of residual urea clearance present in some HEMO patients. In Figure 3, the median percent error in postdialysis serum phosphorus was calculated for each decile of predialysis serum phosphorus, as well as for the lowest vigintile (5% of patients). The top panel shows the error plot for the F4 ‘training set’, and the lower panel shows the error plot for the F36 ‘validation set’. Figure 4 shows very similar data for the 10 predialysis serum phosphorus deciles, this time in terms of measured versus predicted phosphorus reduction ratio.

FIGURE 2.

Percent error in postdialysis serum phosphorus (predicted minus measured) plotted against a variety of potential modifying factors, including size (as postdialysis modeled urea volume), age, predialysis serum phosphorus, dialysis session length and residual kidney urea clearance. Data shown are from patients studied at 4 months of follow-up (F4) in the HEMO Study.

FIGURE 3.

Median percent error in postdialysis serum phosphorus in the HEMO Study at 4 months of follow-up (F4, top panel), or at month 36 (F36, bottom panel). Each decile contains ∼40 patients. The median percent error in the lowest vigintile (5%) of patients in each group is shown as an open circle.

FIGURE 4.

Phosphorus reduction ratio predicted (vertical axis) versus measured (horizontal axis). Shown are the median values for each decile of predialysis serum phosphorus. The median values in the lowest 5% of patients are also shown as an open circle.

Prediction ability in terms of median values for 1-h intradialysis, postdialysis and 30-min postdialysis serum phosphorus values

Figures 5 and 6 expand the analysis to the 1-h intradialysis and 30-min postdialysis sampling points and show the median measured as well as predicted values in the highest decile, the two middle deciles, and the lowest decile, as well as the lowest vigintile. Figure 5 shows the data in the F4 sample and Figure 6 shows the data in the F36 sample.

FIGURE 5.

Predicted versus actual: median intradialysis and postdialysis serum phosphorus values. Training set of patients (HEMO Study at 4 months of follow-up, F4). Actual versus model-predicted median serum phosphorus levels in various quantiles of the HEMO Study patients. The 415 patients in the training set were divided into deciles based on predialysis serum phosphorus level. The highest, middle two (deciles 5 and 6) and lowest deciles are shown, as well as the 20 patients (5%) with the lowest predialysis serum phosphorus values. The symbols and lines used are the same in the four panels. The prediction model was based on the F4 group and the actual measured median phosphorus values shown are also from F4 data. Dialysis sessions are arbitrarily shown ending at 210 min, and the 30-min postdialysis rebound measurement is shown at 240 min. See Table 1 for actual mean dialysis session lengths.

FIGURE 6.

Predicted versus actual: median intradialysis and postdialysis serum phosphorus values. Validation set of patients (HEMO Study at 36 months of follow-up, F36). The same layout is used as in Figure 5, except the measured data are from the F36 ‘validation set’ of patients, whereas the predicted median values are derived from the F4 ‘training set’.

Table 3 shows the serum phosphorus data in greater detail for women and men at both F4 and F36, including the values for percent error at each time point.

Table 3.

Measured and predicted serum phosphorus values and model prediction errors

	F4 group				F36 group				P-value, F4 versus F36
	Women	Men	P-value, women versus men	Both genders	Women	Men	P-value, women versus men	Both genders
No. of cases	247	168		415	210	179		389
Measured serum phosphorus values (mg/dL)
Pre	6.00 (1.68)	5.69 (1.85)	0.078	5.87 (1.76)	5.77 (1.93)	5.82 (1.74)	0.75	5.79 (1.82)	0.41
Intra 60 min	3.61 (1.02)	3.53 (1.16)	0.39	3.58 (1.08)	3.46 (1.10)	3.57 (1.06)	0.31	3.51 (1.09)	0.31
End	2.59 (0.68)	2.73 (0.74)	0.054	2.65 (0.71)	2.49 (0.71)	2.64 (0.69)	0.039	2.56 (0.71)	0.079
End + 30 min	3.22 (0.85)	3.42 (1.10)	0.048	3.30 (0.96)	3.14 (0.90)	3.31 (0.91)	0.067	3.22 (0.91)	0.19
Predicted serum phosphorus values (mg/dL)a
Intra 60 min	3.57 (1.06)	3.51 (1.17)	0.60	3.55 (1.11)	3.47 (1.25)	3.57 (109)	0.44	3.52 (1.18)	0.74
End	2.50 (0.48)	2.64 (0.63)	0.0093	2.55 (0.55)	2.46 (0.62)	2.64 (0.56)	0.0043	2.54 (0.66)	0.76
End + 30 min	3.14 (0.63)	3.36 (0.87)	0.0014	3.23 (0.74)	3.08 (0.76)	3.38 (0.77)	<0.001	3.22 (0.78)	0.91
Prediction errors (percent error)
Intra 60 min %	−0.45 (13)	0.14 (15)	0.67	−0.214 (14)	0.27 (14)	0.72 (14)	0.75	0.48 (14)	0.49
End dialysis %	−0.61 (16)	−0.76 (18)	0.87	−0.608 (17)	1.58 (17)	3.11 (19)	0.40	2.29 (18)	0.019
End + 30 min %	0.46 (17)	1.88 (21)	0.44	1.04 (18)	1.02 (17)	5.73 (21)	0.016	3.67 (19)	0.103

^aThe model iterates to converge on the measured predialysis serum phosphorus value, so the modeled predialysis serum phosphorus values are, for all practical purposes, identical to the measured values.

Error in postdialysis serum phosphorus prediction in nine patients studied at both F4 and F36

The large standard deviation of the percent error in the prediction of intradialysis and postdialysis serum phosphorus values leads one to ask whether there are patient-specific factors that might be associated with a particularly high or low Km value that cannot be predicted by the usual demographic variables. It was impossible to study this question properly using the HEMO patient data, due to the miniscule overlap in patients in the F4 and F36 datasets. Figure 7 shows plots of the percent error in postdialysis serum phosphorus in the nine patients from whom data were available at both time points. As shown, there was no relationship between the percent errors at the two different dialysis sessions.

FIGURE 7.

Percent error in postdialysis serum phosphorus in nine HEMO Study patients who had phosphorus values at both 4 months and 36 months of follow-up (F4 and F36, respectively). The nine patients were sorted by percent error at F4 (solid circles) from lowest to highest and then the values in percent error at F36 were plotted on the same graph (open circles). In the lower panel, the percent error in postdialysis serum phosphorus at F4 (horizontal axis) is plotted against the percent error at F36. There was no correlation.

DISCUSSION

Our results suggest that a modified two-pool model can predict median intradialysis, postdialysis and 30-min postdialysis serum phosphorus concentrations in patients undergoing conventional three times per week hemodialysis regimens. The model was shown to have good results overall in a predialysis serum phosphorus range that represented 95% of values encountered in the HEMO Study. When the predialysis serum level was <3.0 mg/dL (0.97 mmol/L), there was some underestimation of intradialysis and end-dialysis serum phosphorus values, but the amount of underestimation based on median values (e.g. in the lowest 5% of patients) was small.

We were able to mimic the observed behavior of intradialysis and early postdialysis phosphorus by making two modifications to the classical urea kinetic model. First, we increased the size of the distal storage compartment as suggested by Agar et al. [5] and Leypoldt et al. [6], but to only three times the total body water volume. Second, we postulated that the Kc between this storage pool and the extracellular space would not be a fixed value during the entire dialysis session, but rather would vary dynamically during dialysis, increasing substantially if the serum phosphorus fell below an empirically determined threshold level of 3.0 mg/dL (0.97 mmol/l). Below that level, Kc would increase markedly. Either one of these changes applied singly was not sufficient to avoid underestimation of postdialysis serum phosphorus when predialysis values were in the lower range. However, applied together, they were able to mimic the phosphorus profile due to dialysis, even in patients with predialysis phosphorus values in the range of 3.0 mg/dL, the median value of the lowest decile of the HEMO Study patients. Nevertheless, as shown in Figure 4, in those few patients in whom predialysis serum phosphorus values were extremely low (i.e. <3.0 mg/dL), the observed phosphorus reduction ratio due to dialysis not uncommonly was lower than predicted by the model. Thus, in accordance with the observations of Spalding et al. [3], at very low serum phosphorus levels, additional pools of phosphorus are apparently mobilized in an attempt to prevent severe hypophosphatemia, and this behavior is not fully accounted for by the proposed model.

Another method of testing the proposed model was to examine how the weekly predialysis serum phosphorus profile would vary in the course of an asymmetric three times per week dialysis schedule. Although results from different investigators differ somewhat in this area, analysis of FHN Study data and other reports suggest that the phosphorus values measured on Monday/Tuesday are, on average, in the range of 0.2–0.3 mg/dL higher than midweek sessions [11–13], and there is even more of an effect comparing predialysis serum phosphorus levels after 1-day versus 2-day interdialytic intervals [13]. In the proposed model, we examined how the size of the distal storage pool (Vdistal) would affect the within-week variation in predialysis serum phosphorus. We found that the larger the Vdistal, the less the within-week variation. Ratios of Vdistal:Vurea that would still give substantial variation in predialysis serum phosphorus for asymmetric schedules were in the range of 2.0–4.0. We chose a value of 3.0 for the present model. Changing the Vdistal:Vurea ratio also changes the optimal value for Kc slightly. It is possible that higher Vdistal:Vurea ratio values might better fit dialysis schedules where the average session length is longer than 4–5 h. With the Vdistal:Vurea ratio set at 3.0, the optimum value for Kc was 86 mL/min as long as intradialysis serum phosphorus was >3.0 mg/dL (0.97 mmol/L).

The proposed model is agnostic with regard to the exact source of the phosphorus being supplied by the distal storage pool. Some phosphorus may be stored inside cells, as evidenced by the clinical observation of glucose-induced or refeeding hypophosphatemia [15]. It is also possible that some relatively rapid exchange of phosphorus can occur between the intravascular space and bone, as evidenced by the hungry bone syndrome after parathyroidectomy [16]. However, there is no association between alkaline phosphatase or serum parathyroid and intradialysis phosphorus kinetics, as reported recently in the HEMO dataset by Leypoldt et al. [6]. Recent data suggest that an important source of phosphorus removal during dialysis is indeed the intracellular compartment [17].

Our data suggest that for the great majority of patients dialyzed three times per week, the complexity of adding additional pools of phosphorus, as suggested by Spalding et al. [3], or the use of very large storage pools of indefinite size, as suggested by Agar et al. [5] and Leypoldt et al. [6], or even more complex lagged increases in the the release of stored phosphorus, as recently suggested by Debowska Polesczuk et al. [8], may not be required. Over a period of 1 h, postdialysis serum phosphorus values can be predicted with reasonable accuracy, then one has three points of the intradialysis serum phosphorus curve in hand, and the resulting time-averaged level during the dialysis session, which is needed to calculate phosphorus removal, should be calculable with reasonable accuracy, regardless of whether there are minor intradialysis dips and bounces between the data points. We have tested our model in some small datasets of extended (8-h session) dialysis, and the model slightly underestimates the intradialysis serum phosphorus during hours 5–8 of such long sessions. However, the need for adding either additional pools of phosphorus or for postulating a distal pool of infinite size to provide the needed phosphorus may only be required to model very long dialysis sessions, or possibly those long sessions in which the predialysis serum phosphorus levels are quite low.

One problem with any kinetic model of phosphorus removal is the substantial variability seen among patients in the extent to which the serum phosphorus level decreases during dialysis. At the present time, in the absence of multiple repeated sampling in the same patients, it is not clear to what extent such unusual intradialysis phosphorus behavior will replicate from one session to the next, and to what extent there will be regression to more typical mean behavior as evidenced by an analysis of median levels studied in strata of patients grouped by predialysis serum phosphorus or other characteristics. In the present dataset, there were only nine patients who had an analysis of intra- and postdialysis serum phosphorus on two occasions, and although these patients were studied >2 years apart, there was little indication that their deviation from expected average behavior in terms of their serum phosphorus profiles was a patient-specific phenomenon. Thus, whether or not patient-specific correction factors to the values for Kc suggested in the present model might be useful can only be tested in a repeated measures model. It is possible that in some patients, unusually high postdialysis serum phosphorus values are due partly to phosphorus being absorbed from a meal or phosphorus-containing beverages ingested either immediately prior to or during a dialysis session. [18]

One of the limitations of the present dataset is that the HEMO Study did not directly measure phosphate clearance. Dialysate outlet phosphorus was not measured, nor was phosphorus measured in dialysate samples. Great care was taken in the the HEMO Study to get accurate measurements of urea clearance; however, the estimates of phosphorus clearance used in the analysis were based on estimates of in vivo dialyzer urea K₀A values. The estimates for Kc in the present dataset are highly dependent on the estimate of dialyzer phosphorus clearance, as both Kc and dialyzer phosphorus clearance will affect (in opposite directions) the extent of phosphorus reduction during dialysis and the postdialysis serum phosphorus value. With some of the newer dialyzers [19], the ratio of phosphorus to urea dialyzer K₀A may be higher than the 0.40 estimate used in the present calculator equations. In Table 1, we show the calculated phosphorus removal values based on the dialyzer clearance multiplied by the modeled intradialysis serum area under the curve. The values are similar to those reported in studies in which phosphorus recovery was measured in the dialysate.

The potential utility of the present model is that it is simple and diffusion based, and it tracks the concentrations of phosphorus in the proximal and distal pools over the entire week based on estimated dietary phosphorus intake and phosphorus binder dose. This feature potentially allows one to query what equivalent phosphorus binder dose would be needed to maintain a desired predialysis serum phosphorus value prior to any treatment during the week. Alternatively, the model can be used to query what predialysis serum phosphorus value would be achieved in a given patient, as a function of equivalent phosphorus binding dose. Also, the model may be most useful in assessing the changes in serum phosphorus in response to changes in therapy, as discussed in more detail in the Supplementary data. The HTML-javascript code for the model is available in the Supplementary data and is also available at http://www.ureakinetics.org/calculators/batch/phos/phosphatesolver.html (user = solute, password = solver). This HTML page can be downloaded and opened locally on a personal computer with any Web browser. It should be emphasized that the model at present remains preliminary and still requires more validation, especially with regard to the interdialytic period portion, before it can be released for clinical use.