Using Mathematical Algorithms to Modify Glomerular Filtration Rate Estimation Equations

Background The equations provide a rapid and low-cost method of evaluating glomerular filtration rate (GFR). Previous studies indicated that the Modification of Diet in Renal Disease (MDRD), Chronic Kidney Disease-Epidemiology (CKD-EPI) and MacIsaac equations need further modification for application in Chinese population. Thus, this study was designed to modify the three equations, and compare the diagnostic accuracy of the equations modified before and after. Methodology With the use of 99 mTc-DTPA renal dynamic imaging as the reference GFR (rGFR), the MDRD, CKD-EPI and MacIsaac equations were modified by two mathematical algorithms: the hill-climbing and the simulated-annealing algorithms. Results A total of 703 Chinese subjects were recruited, with the average rGFR 77.14±25.93 ml/min. The entire modification process was based on a random sample of 80% of subjects in each GFR level as a training sample set, the rest of 20% of subjects as a validation sample set. After modification, the three equations performed significant improvement in slop, intercept, correlated coefficient, root mean square error (RMSE), total deviation index (TDI), and the proportion of estimated GFR (eGFR) within 10% and 30% deviation of rGFR (P10 and P30). Of the three modified equations, the modified CKD-EPI equation showed the best accuracy. Conclusions Mathematical algorithms could be a considerable tool to modify the GFR equations. Accuracy of all the three modified equations was significantly improved in which the modified CKD-EPI equation could be the optimal one.


Introduction
Chronic kidney disease (CKD) has evolved as a serious challenge to the health and well-being of world population [1,2], among which China has not been spared. The latest incidence of CKD in China is 10.8% [3], equivalent to at least 100 million CKD patients.
Since the development of the Cockcroft-Gault equation in 1976, glomerular filtration rate (GFR) estimation equations have aroused global interests among nephrologists. Among a large number of variations, the Modification of Diet in Renal Disease (MDRD), Chronic Kidney Disease-Epidemiology (CKD-EPI) and MacIsaac equations have been publicly approved and applied [4][5][6][7]. However, ethnicity is one of the essential factors affecting accuracy of the GFR equations [8]. Previous validation studies indicated that modifications are indispensible for superior performances of the GFR equations in Chinese population [9][10][11][12].
Thus, the objective of this study was to create better GFR prediction models for Chinese population, with the first use of mathematical algorithms, due to their specialty at optimizing combinations.

Subjects
All participants in this study signed the informed consent. The participants with severe heart failure, acute renal failure, pleural or abdominal effusion, serious edema or malnutrition, skeletal muscle atrophy, amputation, ketoacidosis were excluded. Patients who recently received glucocorticoid and hemodialysis therapy were also excluded. Nanjing Medical University Ethics committee approved this study.

Laboratory measurements
Serum creatinine (Scr) concentration was assayed by the enzymatic method (Shanghai Kehua Dongling Diagnostic Products Co., Ltd, China) with a reference range of 44,136 mmol/L. Cystatin C concentration was examined by the particle-enhanced immunoturbidimetry method (Beijing Leadman Biomedical Co., Ltd, China) with a reference range of 0.60,1.55 mg/L. Both two markers were examined by an Olympus AU5400 autoanalyzer (Olympus Co., Ltd, Japan).

GFR measurement
A reference GFR (rGFR) was measured by 99 m Tc-DTPA renal dynamic imaging on a single photon emission computed tomography (Siemens E.CAM, Siemens Co., Ltd, Germany) [13]. Participants were informed in advance to have no special change in diet. After height and weight measurement, 300 ml water drinking, and bladder emptying, 185 MBq 99 m Tc-DTPA (purity 95%-99%, Nanjing Senke Co., Ltd, China) was injected into one of the veins of the participant. After images acquisition, rGFR was automatically calculated by a computer with the Gates method [14].

Mathematical modification
The hill-climbing algorithm searched the local optimal solution by fixing each coefficient of the original equations, and orderly adjusting all the coefficients iteratively until no further improvement can be found. To avoid inaccuracy caused by various weights of coefficients, coefficient priorities were switched repeatedly. The simulated annealing algorithm searched the global optimal solution, which remedied the imperfection of the hillclimbing algorithm. Root mean square error (RMSE) was used to    guide the modification process. Matlab software (version R2009a, Math Works Inc., USA) was the platform to accomplish the modification. The entire modification process was based on a random sample of 80% of subjects in each GFR level as a training sample set, the rest of 20% of subjects as a validation sample set.

Statistical Analysis
The Bias was calculated to show mean difference between eGFR and rGFR. Correlated coefficient was calculated using Pearson linear relation analysis to compare the correlation between various eGFR equations and rGFR. Slope and intercept were compared using Bland-Altman analysis. P 10 , P 30 (the percentage of eGFR deviating within 10% and 30% of rGFR) [16][17][18] and the Total Deviation Index (TDI) [19] were also used to compare the accuracy of the equations before and after modification. P value less than 0.05 was taken to consider statistical significance. Statistical analyses were performed using SPSS software, version 16 Table 3.
The modified equations were described in Tables 1 and 2. After modification, the trend to gather around rGFR turned prominent that the extremum or discrete data clearly reduced, and the correlation with rGFR tightened.  Table 4, Fig. 1).
It was obvious that P 10 and P 30 of the modified Scr-equations increased ( Table 4 (Table 4).

Discussion
GFR is the core of CKD evaluation, diagnosis, and classification [20,21]. Due to their simplicity, convenience, and low expense, the equations for GFR evaluation have been extensively applied worldwide [22,23]. Especially, the MDRD and CKD-EPI equations have been successively recommended by Kidney Disease Outcomes Quality Initiative (K/DOQI) and Kidney Disease: Improving Global Outcomes (K/DIGO) [20,24].
Since Scr is a classic biomarker of kidney function, most of the equations during the past 30 years were developed based on it. With the discovery of cystatin C [25], a potentially superior marker [26,27], the equations based on it gradually created and popularized.
However, a great number of previous studies have consistently proved that ''ethnicity'' affects the accuracy of these equations, not only the Scr-based, but also the cystatin C-based [28][29][30]. Our precious studies [12,31,32] indicated that the CKD-EPI and MacIsaac equations could draw eGFR relatively closer to rGFR. Due to the fact that China has the largest and fastest growing number of CKD patients in the world, it is of great significance to establish a more accurate GFR equation for Chinese population.
As powerful optimization capabilities of mathematical algorithms, we firstly introduced the mathematical algorithm to modify the present GFR estimation equations. According to the evidence above, the MDRD, CKD-EPI and MacIsaac equations finally were in selection to accept improvement.
The hill-climbing algorithm, a mathematical optimization technique of the local search family, was first introduced by Goldfeld in 1966 [33]. As an improvement of the depth-first search, the hill-climbing algorithm adopts heuristic strategy, which iteratively searches a better solution by orderly changing one coefficient to the next. However, the hill-climbing algorithm sometimes falls into the local optimization solution rather than the global optimization solution.
The simulated annealing algorithm is an anther artificial intelligence algorithm, which derived from the solid annealing principle. It was put forward by Metropolis in 1953 [34] and then applied into combinatorial optimization field by Kirkpatrick [35]. The simulated annealing algorithm has been widely used in fields such as very large scale integrated circuits, production scheduling, control engineering, machine learning, neural network, and signal processing [36][37][38]. The simulated annealing algorithm, based on iterative solution strategy, is a random optimization algorithm. The simulated annealing algorithm starts with a high initial temperature. Then it randomly searches the global optimization solution of the target function in the solution space with probabilistic jumping property, accompanied by the decline of the temperature parameter to compensate for the drawback of the hill-climbing algorithm.
In this study, the hill-climbing and simulated annealing algorithm substantially increased accuracy of the three selected equations. All the three modified equations performed significant improvement than the originals in slop, intercept, correlated coefficient, RMSE, P 10 , P 30 and TDI. Of the three modified equations, the modified CKD-EPI equation showed the best accuracy.
It is interesting that after modification, improvement of RMSE, P 10 and P 30 in the Scr-based equations (MDRD equation and CKD-EPI equation) were more distinct than that of the cystatin Cbased equation (MacIsaac equation). This fact indicated that Scr could be affected by ethnicity factor easier than cystatin C. Additionally, considering accuracy of the modified MacIsaac equation was similar to that of the modified CKD-EPI equation, plus its simple expression, the modified MacIsaac equation could be also recommended. Another matter should be stated that whether GFR should be adjusted for body surface area is still in debate and confused [39,40]. Therefore, GFR in this study did not make the adjustment. In the end, owing to the inherent unequal distribution of the subjects in each GFR level, accuracy of the original GFR equations varied in different CKD stages [31]. Therefore, to minimize such bias, we modified the equations by stages. It is believed that the modified equations could be better suit for Chinese population.