Improved correction formulas to estimate iohexol clearance from simple models

Measuring glomerular filtration rate by iohexol plasma clearance has become a preferred method in clinical practice and research [1–9], but ambiguities remain regarding the choice of pharmacokinetic modeling when evaluating iohexol concentration. We recently showed that a threecompartment pharmacokinetic model provides better iohexol clearance estimates than the two-compartment approach in elderly subjects in the Berlin Initiative Study (BIS) [7, 8]. However, implementation of the presented model in clinical and research practice is complicated by the required technical expertise and the complexity of Bayesian software [7]. A viable alternative could be the use of correction formulas which rely on traditional oneor two-compartment models and correct the obtained clearance estimates for the bias introduced by the omission of compartments [9–14]. Well-known examples for established correction formulas are the Bröchner-Mortensen (BM) equation [12] and the further simplified Chantler (Ch) formula [14] based on a one-compartment model, which have recently been shown to perform well in iohexol data from the BIS [9]. Such an evaluation remains to be carried out based on a three-compartment model, which is the subject of this letter. Data from the BIS, including 546 individuals with data obtained up to 300 min post-injection, were evaluated [7, 8]. One-, two-, and three-compartment models were used to estimate iohexol clearance (CL1, CL2, and CL3), respectively. CL1 was estimated based on the slow component of iohexol concentrations 120 to 300 min post-dose, CL2 was estimated based on the protocol by Schwartz et al. [11] as carried out in the BIS [8], and CL3 was estimated using the empirical Bayes approach based on the three-compartment model [7]. Equations resembling the BM and Ch formulas were then fitted to correct CL1 and CL2 results, using CL3 as the reference (Table 1) [12, 14]. A leave-one-out cross-validation [15] was utilized to assess the bias and root mean squared error (RMSE), as well as to evaluate Lin’s concordance correlation coefficient (CCC) [16] and the relative total deviation index (TDI) for a range of coverage probabilities (CP) [17]. A TDI ≤ 10% for a CP of 90% (TDI90) and an at least substantial CCC of ≥ 0.95 were considered optimal. R 4.2.1 [18] and NONMEM 7.4.2 [19] were used as statistical software. BMand Ch-like equations performed similarly well (Table 1), with an absolute bias < 1 mL/min, an RMSE between 2.92 and 4.08 mL/min, and a substantial concordance for all equations. The TDI90 goal was achieved with Eqs. 3 and 4 based on two compartments, while it was missed with Eqs. 1 and 2 based on one compartment. Differences between BM-like and Ch-like equations were negligible with equations based on two compartments, while the BM-like equation provided a notably lower absolute bias than the Ch-like equation in the case of one compartment (− 0.0949 versus − 0.357 mL/min). In summary, our evaluation demonstrates that correction formulas in conjunction with oneor two-compartment models can provide adequate clearance estimates with only a minimal loss in accuracy and precision compared to a three-compartment model. When one-compartment estimates are available, the BM-like correction formula might be a good choice, while the simple * Qian Dong qian.dong@uk-koeln.de

Measuring glomerular filtration rate by iohexol plasma clearance has become a preferred method in clinical practice and research [1][2][3][4][5][6][7][8][9], but ambiguities remain regarding the choice of pharmacokinetic modeling when evaluating iohexol concentration. We recently showed that a threecompartment pharmacokinetic model provides better iohexol clearance estimates than the two-compartment approach in elderly subjects in the Berlin Initiative Study (BIS) [7,8]. However, implementation of the presented model in clinical and research practice is complicated by the required technical expertise and the complexity of Bayesian software [7]. A viable alternative could be the use of correction formulas which rely on traditional one-or two-compartment models and correct the obtained clearance estimates for the bias introduced by the omission of compartments [9][10][11][12][13][14]. Well-known examples for established correction formulas are the Bröchner-Mortensen (BM) equation [12] and the further simplified Chantler (Ch) formula [14] based on a one-compartment model, which have recently been shown to perform well in iohexol data from the BIS [9]. Such an evaluation remains to be carried out based on a three-compartment model, which is the subject of this letter.
Data from the BIS, including 546 individuals with data obtained up to 300 min post-injection, were evaluated [7,8]. One-, two-, and three-compartment models were used to estimate iohexol clearance (CL1, CL2, and CL3), respectively. CL1 was estimated based on the slow component of iohexol concentrations 120 to 300 min post-dose, CL2 was estimated based on the protocol by Schwartz et al. [11] as carried out in the BIS [8], and CL3 was estimated using the empirical Bayes approach based on the three-compartment model [7]. Equations resembling the BM and Ch formulas were then fitted to correct CL1 and CL2 results, using CL3 as the reference (Table 1) [12,14]. A leave-one-out cross-validation [15] was utilized to assess the bias and root mean squared error (RMSE), as well as to evaluate Lin's concordance correlation coefficient (CCC) [16] and the relative total deviation index (TDI) for a range of coverage probabilities (CP) [17]. A TDI ≤ 10% for a CP of 90% (TDI 90 ) and an at least substantial CCC of ≥ 0.95 were considered optimal. R 4.2.1 [18] and NONMEM 7.4.2 [19] were used as statistical software.
BM-and Ch-like equations performed similarly well (Table 1), with an absolute bias < 1 mL/min, an RMSE between 2.92 and 4.08 mL/min, and a substantial concordance for all equations. The TDI 90 goal was achieved with Eqs. 3 and 4 based on two compartments, while it was missed with Eqs. 1 and 2 based on one compartment. Differences between BM-like and Ch-like equations were negligible with equations based on two compartments, while the BM-like equation provided a notably lower absolute bias than the Ch-like equation in the case of one compartment (− 0.0949 versus − 0.357 mL/min). In summary, our evaluation demonstrates that correction formulas in conjunction with one-or two-compartment models can provide adequate clearance estimates with only a minimal loss in accuracy and precision compared to a three-compartment model. When one-compartment estimates are available, the BM-like correction formula might be a good choice, while the simple Ch-like formula might be sufficient if two-compartment estimates are available. This provides the means to efficiently estimate iohexol clearance in settings where complexities and costs associated with the implementation of a Bayesian model are prohibitive. Whether the slight loss of accuracy and precision compared to the three-compartment model is acceptable depends on the clinical context and should therefore be judged on a case-by-case basis. Further validation in diverse patient populations is required, and the evaluation of additional, potentially non-linear correction formulas might provide further improvements.

Availability of data and materials
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Competing interests
The authors declare no competing interests.

Disclaimer
The funder had no role in design, collection, analysis, and interpretation of data, and writing and publication of the manuscript.
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