Solute-Solver: A Web-Based Tool for Modeling Urea Kinetics for a Broad Range of Hemodialysis Schedules in Multiple Patients

Urea Modeling Approach 

The urea modeling approach used in this program is similar to that used to analyze data from the National Institutes of Health Hemodialysis (HEMO) clinical study9, 10 and that is currently used to analyze solute clearances in the Frequent Hemodialysis Network (FHN) trials of daytime and nocturnal dialysis delivered 6 times weekly.11 The equations used for the 1- and 2-pool models, both of which allow a variable volume, are modified from several sources.3, 9, 10 Urea generation rate (G) is calculated by using a previously described iterative method12 that systematically adjusts G while cycling until the modeled predialysis BUN matches the measured value.

Dialyzer Clearance 

Dialyzer clearance is resolved in 1 of 3 ways. In default mode, diffusive dialyzer clearance (Kdif) is calculated from the dialyzer K0A and countercurrent flow rates of dialysate and blood (Qb and Qd), assuming an exponential change in concentrations across the dialyzer from inflow to outflow port.3 In dialyzer clearance (Kd) input mode, the estimated dialyzer clearance is entered directly. In V input mode, the estimated urea distribution volume is entered and Kd is estimated by using urea kinetic equations modified from a method described by Casino et al,13 the details of which are beyond the scope of the present report. The dialysis-machine-reported conductivity-based clearance (K) multiplied by time (t), or Kt, can also be entered, from which the program also can compute Kd.

In default mode, the first step is to adjust the user-supplied K0A published by the dialyzer manufacturer. K0A values supplied by the dialyzer manufacturer most often are derived from in vitro measurements of solute removal from saline solutions; thus, the program adjusts these downward to more realistic in vivo values.14 The equation used to estimate in vivo K0A from the industry-supplied value (equation 1; all numbered equations are provided in the Appendix) was designed to be consistent with simultaneous inlet-outlet measurements of dialyzer clearance during the HEMO Study.14 The program also adjusts the dialyzer K0A as a function of the dialysate flow rate. At greater dialysate flow rates, better permeation of dialysate into the fiber bundle effectively increases the dialyzer surface area and K0A.14, 15 Low dialysate flow rates (<500 mL/min) have the opposite effect, which may significantly decrease K0A (even to levels below those calculated by the presently used equation).16, 17 The diffusive dialyzer clearance (Kdif) then is determined from the estimated in vivo K0A value, Qb, and Qd according to equation 2.3 Reduction from whole blood to blood water clearance in the clearance equation uses a multiplier of 0.86 for blood water content. This is a weighted average of the presumed urea distribution space in erythrocytes (0.72) and plasma (0.93), assuming hematocrit of 0.33.3 Kdif calculated in this manner then is adjusted for the expected fluid removal rate during each dialysis session by using equation 3.3 The resulting clearance is a total dialyzer clearance (Kd) that includes the ultrafiltration component. When hemodiafiltration is used, the program computes the additional clearance of urea due to this process based on the predilution and/or postdilution replacement fluid infusion rate.18

Volume Changes and Ultrafiltration 

Assumptions are made that fluid accumulation occurs at a constant rate during the week excluding dialysis sessions, and fluid removal during each dialysis session is adjusted by dialysis staff to achieve the same postdialysis weight. The duration of each dialysis treatment during the week is assumed to be identical; therefore, sessions after longer interdialysis intervals require a greater rate of ultrafiltration. This results in a slightly greater Kd for sessions that follow longer interdialysis intervals.

Single-Pool Analysis 

Single-pool (1p) analysis is not strictly required, but is computed by the program for comparison purposes and to meet regulatory requirements and because single-pool analysis yields values for modeled urea distribution volume (V1p) and generation rate (G1p) that subsequently can be adjusted to estimated 2-pool values that are close to the final values, allowing for fewer iterations in the 2-pool (2p) portion of the program that requires numerical analysis for integration (discussed later). The variable volume single-pool equations described by Gotch are used.3, 19 Briefly, an initial value for G1p is estimated. Using the estimated total dialyzer clearance Kd, the predialysis and postdialysis BUN, and the fluid removal rate, a value for V1p is computed. Next, using single-pool kinetic equations and these initial estimates of V1p and G1p, a BUN profile is projected for an entire week. The projected predialysis BUN value 7 days later then is compared with the measured value. If these do not match, the value for G1p is adjusted upward or downward and the weekly urea profile projection is repeated, attempting again to match the measured value.12 Because changes in G1p will affect the calculation of V1p, after each adjustment of G1p, the value for V1p is recalculated. When G1p has stabilized and the BUN projection after 7 days matches the predialysis value, values for G1p and V1p are accepted and the single-pool Kt/V (spKt/V) is computed as Kd • Td/V1p.

Two-Pool Analysis 

Unlike the 1-pool model, the differential equations describing the 2-pool model have no explicit solutions. It therefore is necessary to use other methods, such as numerical analysis, to find approximate values for V2p and G2p. The initial value for extracellular volume (Ve) is assumed to be one-third of an estimated 2-pool value for V plus the volume of fluid removed during the modeled dialysis session. This initial approximate estimate of V2p is computed from the modeled V1p, adjusted to a 2-pool volume based on the estimated equilibrated postdialysis BUN as described by Daugirdas and Smye.20 The starting value for G2p is 0.95 • G1p. Using these starting values, the “blood-draw-day” dialysis session is modeled. The program begins with the measured predialysis BUN value. This BUN concentration initially is assigned to both the extracellular (Ce) and intracellular spaces (Ci). The program calculates Ce and Ci, as well as Ve, as dialysis progresses by applying small time increments to the differential equations that describe changes in Ce, Ci, and Ve caused by movement of solute between compartments and both solute and water across the dialyzer membrane. Two differential equations describing urea mass balance (equations 4a and 4b) form the root of the 2-compartment analysis. The integration interval is 1 minute during the first 50 minutes of dialysis and the last 5 to 10 minutes of dialysis, but expands to 10 minutes between these time ranges. Integration is facilitated by using a fourth-order Runge-Kutta numerical method.21

At the end of the blood-draw-day dialysis session, the calculated Ce is compared with the measured postdialysis value. The initial estimate of V2p then is adjusted in the appropriate direction (increased if modeled postdialysis Ce is less than the measured postdialysis BUN and decreased if modeled postdialysis Ce is greater than the measured postdialysis BUN). The blood-draw-day dialysis integration is repeated with V2p adjustment multiple times until the computed postdialysis Ce value matches the measured postdialysis BUN.

The fitted value for V (V2p) then is applied together with the estimated starting value for G2p to project a weekly BUN profile using the numerical method described. A 1-minute integration interval is used for the first 50 minutes postdialysis, and this is lengthened to 10 minutes until 5 minutes before the next dialysis session, a maneuver that substantially shortens the time required to complete the weekly BUN projection. Then, as in the single-pool simulation, the value of Ce predialysis 7 days later is compared with the measured predialysis value. G2p then is adjusted in the appropriate direction. Because the value for G2P will affect the value for V2p, the blood-draw-day dialysis session is remodeled after each adjustment to G2p, recomputing a new value for V2p, and then a new weekly projection of Ce and Ci is performed. Using the 2-pool model, Ce and Ci remain slightly different throughout the week, even predialysis. This difference is important for stability of the calculations, especially for long nocturnal dialysis, in which low BUN values and relatively short interdialysis intervals often result in nontrivial differences between predialysis Ce and Ci. When the 7-day projected predialysis Ce matches the measured predialysis value, the value for G2p is accepted along with the most recent value for V2p, and the 2-pool model is complete. A typical plot of the final BUN concentration profile is shown in Fig 1.

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Figure 1. Plot of predicted weekly blood urea nitrogen (BUN) concentration. Calculations are similar to the process used for urea kinetic modeling in Solute-Solver. The upper horizontal dotted line shows mean predialysis BUN level, and the lower horizontal dashed line shows time-averaged concentration. All concentrations have been reduced from pure aqueous values by multiplying by 0.93 (to correspond to plasma values measured during the blood-draw-day dialysis session).

Residual Renal Clearance 

For calculation of residual renal clearance (Kru), the program computes the time-averaged BUN level during the collection period from the computed weekly BUN profile. The program is configured to calculate Kru when the collection period begins and ends within the interdialysis interval immediately preceding the modeled dialysis session. An iterative solution is used because Kru contributes to the weekly urea nitrogen profile. Because a 2-pool analysis is required for proper determination of BUN values during the collection period, in patients with Kru, when Kru is computed from a urine collection, the single-pool loop is repeated with the final value of Kru to ensure an accurate single-pool Kt/V.

Equilibrated Kt/V 

Equilibrated Kt/V (eqKt/V) is defined here as Keq • Td/V2p. Keq is defined as the patient “equilibrated” clearance. This is calculated solving the single-pool model equations for K, but using the whole-body BUN concentration (Cwb) values predialysis and postdialysis (Cwb_pre and Cwb_post, respectively) along with the other values listed in equation 5. The single-pool model does not have an explicit solution for K because K appears on both sides of equation 5.3 This equation therefore is solved by using iteration. Initially, the single-pool value for K, Kd, is used as the estimate for Keq (equilibrated clearance) and is placed in the right-hand side of equation 5. The resulting value of Keq will be lower than the original estimate, with the true answer lying somewhere in between, resolved by using iteration.

Two-Pool Kt/V 

eKt/V is not a “double-pool” Kt/V. The latter would be computed as Kd • Td/V2p. This is computed internally in the program, but is not reported in the default version of the program. If desired, one can easily reconfigure the code to add this to the output reports.

Protein Catabolic Rate 

Protein catabolic rate (PCR) is calculated from G2p normalized to the 2-pool volume by using an equation (equation 6) described by Gotch.3, 18 This equation converts V2p to an ideal or “normalized” body weight (assuming 58% body water content) for use as the PCR denominator.

Time-Averaged BUN: Extracellular and Whole-Body Average 

Values for Ce, Ci, and Ve determined at simultaneous intervals throughout the week are stored as an array in computer memory. Whole-body urea nitrogen concentrations (Cwb) at each time also are computed and stored as the average of Ce and Ci weighted by the relative values of Ve and Vi. The program then computes the time-averaged urea nitrogen concentrations TACe and TACwb from areas under the Ce and Cwb curves, respectively.

Continuous Clearances 

Two continuous clearances can be calculated: G/TACwb22 and G/(mean Cwb_pre), the latter known as “standard” Kt/V.23, 24, 25 These clearances, expressed in mL/min, then are normalized to V2p and expressed on a weekly basis as a fractional volume cleared per week, analogous to Kt/V. In the default version, the only clearances reported are different versions of the “standard” Kt/V, G/mean Cwb_pre, but G/TACwb can be easily added to the output report.

Treatment or “Dialysis Standard” Kt/V 

This measure isolates the clearance due to dialysis treatment and excludes effects of residual renal function. It is computed by simulating a weekly urea profile for the patient in question by assuming that Kru is zero, but V2p and G2p are unchanged. After urea concentrations are generated (which are now higher due to the lack of Kru), the clearance is calculated as G/(mean Cwb_pre) multiplied by 10,080/V2p, using the new value for mean predialysis urea nitrogen concentration.

Surface Area Normalized Clearances 

Surface area (S), anthropometric V (Vant), and the ratio of Vant/S are computed from optional user input of demographic variables. To convert to surface area normalization, the continuous clearances described previously are multiplied by (Vant/S)/20.26 The average of the median Vant/S ratios in male and female dialysis patients in the HEMO baseline data set was close to 20, and V/S ratio is higher in large patients and men than in small patients and women. Multiplying any of the versions of “standard” Kt/V described by (Vant/S)/20 will adjust that particular “standard” Kt/V value for surface area because it adjusts for the ratio of Vant/S versus the population median that was used to derive overall adequacy standards.26 Surface area normalized clearances should be interpreted with caution in patients with significant amputations, as well in pediatric patients in which the potential impact of V/S ratio has not been studied.