Using Mathematical Algorithms to Modify Glomerular Filtration Rate Estimation Equations
et al. (2013) Using Mathematical Algorithms to Modify Glomerular Filtration Rate Estimation Equations. PLoS
ONE 8(3): e57852. doi:10.1371/journal.pone.0057852
Using Mathematical Algorithms to Modify Glomerular Filtration Rate Estimation Equations
Xiaohua Pei 0
Wanyuan Yang 0
Shengnan Wang 0
Bei Zhu 0
Jianqing Wu 0
Jin Zhu 0
Weihong Zhao 0
Giuseppe Remuzzi, Mario Negri Institute for Pharmacological Research and Azienda Ospedaliera Ospedali Riuniti di Bergamo, Italy
0 1 Division of Nephrology, Department of Geriatrics, The First Affiliated Hospital of Nanjing Medical University , Nanjing, Jiangsu , China , 2 Institute of Pattern Recognition and Machine Intelligence, School of Computer Science, Nanjing University of Science and Technology , Nanjing, Jiangsu , China , 3 Division of Respiration, Department of Geriatrics, The First Affiliated Hospital of Nanjing Medical University , Nanjing, Jiangsu , China
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 77.14625.93 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.
Funding: This work was supported by the Innovation of Science and Technology Achievement Transformation Fund of Jiangsu Province BL2012066, the National
Natural Science Foundation of China H0511-810705, and the grants from the Major State Basic Research Development Program of China 2013CB530803, a Project
Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions JX10231801. The funders had no role in study design, data
collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
. These authors contributed equally to this work.
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% , equivalent to at least 100 million
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 .
However, ethnicity is one of the essential factors affecting accuracy
of the GFR equations . Previous validation studies indicated
that modifications are indispensible for superior performances of
the GFR equations in Chinese population .
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
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.
Table 1. Equations before and after modification by mathematical algorithms.
GFR = (86.7/cystatin C)24.2
GFR = (77.30/cystatin C)+2.32
MDRD: Modification of Diet in Renal Disease; GFR: glomerular filtration rate.
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).
A reference GFR (rGFR) was measured by 99 mTc-DTPA renal
dynamic imaging on a single photon emission computed
tomography (Siemens E.CAM, Siemens Co., Ltd, Germany)
. 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 mTc-DTPA
GFR = 1446(Scr/
GFR = 1446(Scr/
GFR = 1416(Scr/
GFR = 1416(Scr/
GFR = 1446(Scr/
GFR = 1446(Scr/
GFR = 1416(Scr/
GFR = 1416(Scr/
(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
The estimation equations, including the MDRD, CKD-EPI and
MacIsaac equations , were shown in Tables 1 and 2.
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
Cystatin C (mg/L)
Serum creatinine (mmol/L)
Blood urea nitrogen (mmol/L)
Uric acid (mmol/L)
Fasting blood glucose (mmol/L)
Low density lipoprotein (mmol/L)
GFR $90 ml/min
GFR 6089 ml/min
GFR 3059 ml/min
GFR ,30 ml/min
History of subjects
Diabetic kidney disease
Renal artery stenosis
Indeterminate etiology or unknown 106
Number of subjects (n)
Number of subjects (n)
GFR: glomerular filtration rate, measured by 99 mTc-DTPA renal dynamic
imaging; ADPKD: adult dominant polycystic kidney disease; values are means 6
CKD-EPI: Chronic Kidney Disease Epidemiology Collaboration; GFR: glomerular
A single for the CKD-EPI equation is GFR = 1416min (Scr/k, 1) a6max
(Scr/k, 1)21.05760.993 age (61.018, if female), where Scr is serum creatinine, k is
0.7 for females and 0.9 for males, a is 0.156 for females and 0.074 for males, min
indicates the minimum of Scr/k or 1, and max indicates the maximum of Scr/k
Mean difference (ml/min)
Root mean square error
Total deviation index 70%
Total deviation index 75%
Total deviation index 80%
MDRD: modification of diet in renal disease; CKD-EPI: Chronic Kidney Disease
Epidemiology Collaboration; Relationship: correlated coefficient of estimated
GFR (eGFR) with reference GFR (rGFR); P10: proportion of eGFR that were within
10% of rGFR; P30: proportion of eGFR that were within 30% of rGFR.
**P,0.01, compared with the original equation.
guide the modification process. Matlab software (version R2009a,
Math Works Inc., USA) was the platform to accomplish the
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.
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. P10, P30 (the
percentage of eGFR deviating within 10% and 30% of rGFR)
 and the Total Deviation Index (TDI)  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 16.0 (SPSS Inc., Chicago, USA) and
Medcalc for Windows, (version 184.108.40.206, Medcalc Software,
A total of 703 Chinese subjects, including 422 males and 281
females, were recruited in this study, who attended The First
Affiliated Hospital of Nanjing Medical University between
December 2009 and October 2012. The subjects aged 1895 yr,
mean 52.38616.86 yr, with the average rGFR, cystatin C and Scr
77.14625.93 ml/min, 1.3560.78 mg/L and
105.00683.31 mmol/L, respectively. The GFR levels, medical
history, detailed clinical characteristics of subjects were listed in
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. The correlated coefficients of
eGFR from the MDRD, CKD-EPI and MacIsaac equations rise
from 0.784, 0.846 and 0.777 to 0.804, 0.851 and 0.810,
respectively). Mean difference of the MDRD and CKD-EPI got
smaller (MDRD: 7.42 ml/min decreased to 24.84 ml/min,
CKD-EPI: 2.38 ml/min to 2.17 ml/min), but mean difference
of the MacIsaac increased (23.1 ml/min to 25.30 ml/min).
Intercept and slope of eGFR from the MDRD, CKD-EPI and
MacIsaac equations became narrowed in Bland-Altman analysis
(intercept: 226.70, 214.26, 220.59 to 14.11, 28.91, 212.53,
slope: 0.42, 0.22, 0.25 to 0.14, 0.15, 0.11, respectively) (Table 4,
Meanwhile, after modification, P10 of the MDRD, CKD-EPI
and MacIsaac equations increased from 30.7%, 32.3%, 36.1% to
35.7%, 38.4%, 37.3%, synchronously, P30 increased from 75.7%,
79.4%, 82.4% to 84.1%, 84.5% and 85.3%. Another, TDI
(including TDI 70%80%) were also significantly decreased
It was obvious that P10 and P30 of the modified Scr-equations
increased (Table 4, Fig. 2). Compared with the modified cystatin
C-based equation (MacIsaac equation), RMSE of the modified
two Scr-based equations (MDRD and CKD-EPI equation) were
decreased in sharp contrast (Table 4).
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 , a potentially superior
marker [26,27], the equations based on it gradually created and
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 . 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 . 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  and then
applied into combinatorial optimization field by Kirkpatrick .
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 . 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
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, P10, P30 and TDI. Of the three modified
equations, the modified CKD-EPI equation showed the best
It is interesting that after modification, improvement of RMSE,
P10 and P30 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 .
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.
We thank Jianfeng Ma and Chengjing Yan for laboratory measurements,
and Lihua Bao and Zhaoqiang Xu for GFR measurement. We appreciate
the two Reviewers for their valuable and constructive comments, which
opened and enlightened our mind.
Conceived and designed the experiments: WHZ XHP JZ. Performed the
experiments: XHP WYY SNW BZ JQW JZ. Analyzed the data: XHP
WYY SNW JZ. Contributed reagents/materials/analysis tools: XHP JZ.
Wrote the paper: XHP.
1. Wen CP , Cheng TY , Tsai MK , Chang YC , Chan HT , et al. ( 2008 ) All-cause mortality attributable to chronic kidney disease: a prospective cohort study based on 462 293 adults in Taiwan . Lancet 371 : 2173 - 2182 .
2. Glassock RJ , Winearls C ( 2008 ) An epidemic of chronic kidney disease: fact or fiction? Nephrol Dial Transplant 23 : 1117 - 1121 .
3. Zhang L , Wang F , Wang L , Wang W , Liu B , et al. ( 2012 ) Prevalence of chronic kidney disease in China: a cross-sectional survey . Lancet 379 : 815 - 822 .
4. Levey AS , Coresh J , Balk E , Kausz AT , Levin A , et al. ( 2003 ) National Kidney Foundation practice guidelines for chronic kidney disease: evaluation, classification, and stratification . Ann intern med 139 : 137 - 147 .
5. Levey AS , Stevens LA ( 2010 ) Estimating GFR using the CKD epidemiology collaboration (CKD-EPI) creatinine equation: more accurate GFR estimates, lower CKD prevalence estimates, and better risk predictions . Am J Kidney Dis 55 : 622 - 627 .
6. Stevens LA , Li S , Kurella Tamura M , Chen SC , Vassalotti JA , et al. ( 2011 ) Comparison of the CKD Epidemiology Collaboration (CKD-EPI) and Modification of Diet in Renal Disease (MDRD) study equations: risk factors for and complications of CKD and mortality in the Kidney Early Evaluation Program (KEEP) . Am J Kidney Dis 57 : S9 - 16 .
7. Chudleigh RA , Ollerton RL , Dunseath G , Peter R , Harvey JN , et al. ( 2009 ) Use of cystatin C-based estimations of glomerular filtration rate in patients with type 2 diabetes . Diabetologia 52 : 1274 - 1278 .
8. Rule AD , Teo BW ( 2009 ) GFR estimation in Japan and China: what accounts for the difference ? Am J Kidney Dis 53 : 932 - 935 .
9. Zuo L , Ma YC , Zhou YH , Wang M , Xu GB , et al. ( 2005 ) Application of GFRestimating equations in Chinese patients with chronic kidney disease . Am J Kidney Dis 45 : 463 - 472 .
10. Du X , Hu B , Jiang L , Wan X , Fan L , et al. ( 2011 ) Implication of CKD-EPI equation to estimate glomerular filtration rate in Chinese patients with chronic kidney disease . Ren Fail 33 : 859 - 865 .
11. Liu X , Lv L , Wang C , Shi C , Cheng C , et al. ( 2010 ) Comparison of prediction equations to estimate glomerular filtration rate in Chinese patients with chronic kidney disease . Intern Med J 42 : e59 - 67 .
12. Pei XH , He J , Liu Q , Zhu B , Bao LH , et al. ( 2012 ) Evaluation of serum creatinine- and cystatin C-based equations for the estimation of glomerular filtration rate in a Chinese population . Scand J Urol Nephrol 46 : 223 - 231 .
13. Heikkinen JO , Kuikka JT , Ahonen AK , Rautio PJ ( 2001 ) Quality of dynamic radionuclide renal imaging: multicentre evaluation using a functional renal phantom . Nucl Med Commun 22 : 987 - 995 .
14. Gates G ( 1984 ) Computation of glomerular filtration rate with Tc-99 m DTPA: an in-house computer program . J Nucl Med 25 : 613 - 618 .
15. Levey AS , Bosch JP , Lewis JB , Greene T , Rogers N , et al. ( 1999 ) A more accurate method to estimate glomerular filtration rate from serum creatinine: a new prediction equation . Ann Intern Med 130 : 461 - 470 .
16. Levey AS , Stevens LA , Schmid CH , Zhang YL , Castro AF , et al. ( 2009 ) A new equation to estimate glomerular filtration rate . Ann Intern Med 150 : 604 - 612 .
17. Macisaac RJ , Tsalamandris C , Thomas MC , Premaratne E , Panagiotopoulos S , et al. ( 2006 ) Estimating glomerular filtration rate in diabetes: a comparison of cystatin-C- and creatinine-based methods . Diabetologia 49 : 1686 - 1689 .
18. Coresh J , Stevens L ( 2006 ) Kidney function estimating equations: where do we stand? Curr Opin Nephrol Hypertens 15 : 276 - 284 .
19. Geo`rgia E , Carlos A , Josep C ( 2010 ) The Total Deviation Index estimated by Tolerance Intervals to evaluate the concordance of measurement devices . BMC Med Res Methodol.
20. National Kidney Foundation . ( 2002 ) K/DOQI clinical practice guidelines for chronic kidney disease: evaluation, classification, and stratification . Am J Kidney Dis 39 : S1 - 266 .
21. Levey AS , Eckardt KU , Tsukamoto Y , Levin A , Coresh J , et al. ( 2005 ) Definition and classification of chronic kidney disease: a position statement from Kidney Disease: Improving Global Outcomes (KDIGO) . Kidney Int 67 : 2089 - 2100 .
22. Gansevoort RT , Matsushita K , van der Velde M , Astor BC , Woodward M , et al. ( 2011 ) Lower estimated GFR and higher albuminuria are associated with adverse kidney outcomes . Kidney Int 80 : 93 - 104 .
23. Zhang R , Zheng L , Sun Z , Zhang X , Li J , et al. ( 2011 ) Decreased Glomerular Filtration Rate Is Associated with Mortality and Cardiovascular Events in Patients with Hypertension: A Prospective Study . PLoS One 6 : e27359 .
24. Levey AS , de Jong PE , Coresh J , El Nahas M , Astor BC , et al. ( 2011 ) The definition, classification, and prognosis of chronic kidney disease: a KDIGO Controversies Conference report . Kidney Int 80 : 17 - 28 .
25. Grubb A , Simonsen O , Sturfelt G , Truedsson L , Thysell H ( 1985 ) Serum concentration of cystatin C, factor D and beta 2-microglobulin as a measure of glomerular filtration rate . Acta Med Scand 218 : 499 - 503 .
26. Jaisuresh K , Sharma RK , Mehrothra S , Kaul A , Badauria DS , et al. ( 2012 ) Cystatin C as a Marker of Glomerular Filtration Rate in Voluntary Kidney Donors . Exp Clin Transplant 10 : 14 - 17 .
27. Peralta CA , Katz R , Sarnak MJ , Ix J , Fried LF , et al. ( 2011 ) Cystatin C identifies chronic kidney disease patients at higher risk for complications . J Am Soc Nephrol 22 : 147 - 155 .
28. Leung TK , Luk AO , So WY , Lo MK , Chan JC ( 2010 ) Development and validation of equations estimating glomerular filtration rates in Chinese patients with type 2 diabetes . Kidney Int 77 : 729 - 735 .
29. Earley A , Miskulin D , Lamb EJ , Levey AS , Uhlig K ( 2012 ) Estimating Equations for Glomerular Filtration Rate in the Era of Creatinine Standardization: A Systematic Review . Ann Intern Med.
30. Stevens LA , Padala S , Levey AS ( 2010 ) Advances in glomerular filtration rateestimating equations . Curr Opin Nephrol Hypertens 19 : 298 - 307 .
31. Pei XH , Bao LH , Xu ZQ , Yan C , He J , et al. ( 2012 ) Diagnostic accuracy of cystatin C and glomerular filtration rate formulae in Chinese non-elderly and elderly population . J Nephrol.
32. Pei XH , Liu Q , He J , Bao LH , Yan CJ , et al. ( 2012 ) Are cystatin C-based equations superior to serum creatinine-based equations in elderly . Chinese Int Urol Nephrol 44 : 1877 - 1884 .
33. Goldfeld SM , Quandt RE , Trotter HF ( 1966 ) Maximization by quadratic hillclimbing . Econometrica : 541 - 551 .
34. Metropolis N , Rosenbluth A , Rosenbluth M , Teller A , Teller E ( 1953 ) Simulated annealing . J Chem Phys 21 : 1087 - 1092 .
35. Kirkpatrick S , Gelatt CD Jr, Vecchi MP ( 1983 ) Optimization by simulated annealing . Science 220 : 671 - 680 .
36. Anand S , Saravanasankar S , Subbaraj P ( 2012 ) A multiobjective optimization tool for Very Large Scale Integrated nonslicing floorplanning . Int J Circ Theo Appl.
37. Ribas I , Leisten R , Framin an JM ( 2010 ) Review and classification of hybrid flow shop scheduling problems from a production system and a solutions procedure perspective . Comput Oper Res 37 : 1439 - 1454 .
38. Mellit A , Kalogirou SA , Hontoria L , Shaari S ( 2009 ) Artificial intelligence techniques for sizing photovoltaic systems: A review . Renew Sust Energ Rev 13 : 406 - 419 .
39. Geddes CC , Woo YM , Brady S ( 2008 ) Glomerular filtration rate-what is the rationale and justification of normalizing GFR for body surface area? Nephrol Dial Transplant 23 : 4 - 6 .
40. Delanaye P , Radermecker RP , Rorive M , Depas G , Krzesinski JM ( 2005 ) Indexing glomerular filtration rate for body surface area in obese patients is misleading: concept and example . Nephrol Dial Transplant 20 : 2024 - 2028 .