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UKPDS58—modeling glucose exposure as a risk factor for photocoagulation in type 2 diabetes
Journal of Diabetes and Its Complications 16 (2002) 371 – 376
UKPDS58—modeling glucose exposure as a risk factor for photocoagulation in type 2 diabetes Richard J. Stevens*, Irene M. Stratton, Rury R. Holman Diabetes Trials Unit, Oxford Centre for Diabetes, Endocrinology and Metabolism, University of Oxford, Oxford, UK Received 4 October 2001; received in revised form 18 December 2001; accepted 28 December 2001
Abstract In type 2 diabetes, the risk of retinopathy, and of retinal photocoagulation, rises with time after diagnosis of diabetes. In this paper, mathematical modeling shows that this ageing effect is attributable to the rise in glycemia with time since diagnosis of diabetes. Mathematical models were fitted to data from 3648 patients from the UK Prospective Diabetes Study (UKPDS). A proportional hazards model, in which time and glycemia measured by HbA1c are independent risk factors for photocoagulation, was compared to a model in which time does not contribute except through a measure of cumulative glucose exposure. Since likelihood ratio tests cannot be applied to non-nested models, graphical methods were used to compare the two models. The glucose exposure model was able to fit variation in survival with time at least as well as the proportional hazards model. The proportional hazards model, however, seriously underestimates the differences in two groups of different mean HbA1c. We conclude that duration of diabetes and HbA1c level better predict risk for photocoagulation when treated as two components of cumulative glucose exposure, than when treated as independent risk factors. D 2002 Elsevier Science Inc. All rights reserved. Keywords: Modeling; Glucose exposure; Duration of diabetes; Age; Retinal photocoagulation
1. Introduction Glycated hemoglobin, and its subfraction HbA1c, provide integrated measures of the degree of hyperglycemia in an individual over the preceding 2 –3 months. Previous studies have found that an elevated HbA1c or glycated hemoglobin indicates an increased risk of diabetic retinopathy (Klein, Klein, & Moss, 1995; Nakagami, Kawahara, Hori, & Omori, 1997). Randomized controlled clinical trials in both type 1 and type 2 diabetic subjects have demonstrated that improved glycemic control can reduce risk of progression of retinopathy, suggesting a causative relationship between glycemia and retinopathy (DCCT Research Group, 1995; Ohkubo et al., 1995; UKPDS Group, 1998a). It is also known that the risk for retinal photocoagulation rises with duration of diabetes in both type 1 and type 2 (Klein et al., 1995; UKPDS Group, 1997), and that HbA1c levels tend to rise with duration of diabetes (Shorr et al., 2000; UKPDS Group, 1999). In type 2 diabetes, duration of * Corresponding author. Diabetes Trials Unit, Radcliffe Infirmary, 2nd Floor, Woodstock Road, Block 11, Oxford OX2 6HE, UK. Tel.: +44-1865224463; fax: +44-1865-723884. E-mail address: [email protected] (R.J. Stevens).
diabetes is a risk factor for retinopathy, but age at diagnosis of diabetes is not (UKPDS Group, 1997). Thus, in type 2 diabetes, age is a risk factor only as a marker for duration of diabetes. Given that both glycemia and duration of diabetes are risk factors for retinopathy and for retinal photocoagulation, one possibility is that these effects are independent. Another is that the increased risk over time is attributable in whole or large part to cumulative exposure to high levels of HbA1c over time (Dowse et al., 1998). In type 1 diabetes, the latter hypothesis is supported by evidence from The Diabetes Control and Complications Trials (DCCT), in which patients with the lowest HbA1c levels had no increase in risk for retinopathy progression over time (DCCT Research Group, 1995). The present paper explores the same hypothesis for type 2 diabetic patients, using data from the UK Prospective Diabetes Study (UKPDS), through the use of statistical models. The relationship between HbA1c and risk for retinal photocoagulation has been reported in a previous UKPDS paper (UKPDS Group, 2000a). The aim of the present paper is to use statistical models to explore two alternative hypotheses, not to present a fully generalizable risk model. In the first hypothesis, duration of diagnosed
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diabetes, and recent HbA1c, are risk factors that act independently and multiplicatively, as in a proportional hazards model. In the second, cumulative exposure to high levels of HbA1c acts directly on risk for photocoagulation, and duration of diabetes does not affect risk directly but through increased cumulative HbA1c exposure. Each of these can be represented by a parametric model, and the fit of each model to the observed data can be evaluated.
2. Methods
HbA1c was measured by DCCT-aligned (UKPDS Group, 1998a) HPLC (Biorad Diamat Automated Glycosylated Haemoglobin Analyser), non-diabetic range 4.5– 6.2% (Cull et al., 1997; UKPDS Group, 1994). There were 3648 patients, with both baseline retinopathy grade and follow-up HbA1c data, included in this analysis. Patients with photocoagulation prior to diagnosis of diabetes were excluded by the UKPDS protocol. Table 1 shows baseline characteristics of the patients. For patients with incomplete HbA1c data, linear regression was used to interpolate and extrapolate missing values. This was necessary for 1768 patients, for a total of 4448 values.
2.1. Patients
2.2. Endpoint
The UKPDS recruited 5102 patients with newly diagnosed type 2 diabetes, in 23 centres, between 1977 and 1991, and continued until the study closed on 30th September 1997. In addition to clinician diagnosis of diabetes, patients were included in the study if their fasting plasma glucose exceeded 6 mmol/L on two further occasions. Exclusion criteria were severe vascular disease, accelerated hypertension, severe retinopathy requiring photocoagulation, renal failure (serume creatinine > 175 mmol/L), life threatening disease, disease requiring steroid treatment, or occupation precluding treatment with insulin. The UKPDS protocol has been described in detail elsewhere (UKPDS Group, 1991, 1996). The study design conformed to the Declarations of Helsinki (1975 and 1983), and was approved by the local Research Ethics Committee in each centre. Each patient gave informed, witnessed consent. Patients in 22 centres recruited since 1983 had retinopathy assessed by four-field retinal photography on study entry and every 3 years thereafter. At each assessment, a grade was assigned according to a modification of the ETDRS scale (ETDRS Research Group, 1991). Details of the photographic methods have been described elsewhere (UKPDS Group, 1998b). For this analysis, we define indicator variables B35 and B43 for modified retinopathy grade in the worse eye at study entry, as follows:
The decision for photocoagulation was taken by opthalmologists independent of the UKPDS (UKPDS Group, 1998a). We include both pan-retinal photocoagulation, and focal photocoagulation for macular edema.
define B35 = 1 for modified ETDRS grade 35 (microaneurysms plus minimal haemorrhages and/or hard exudates), otherwise, B35 = 0; and B43 = 1 for modified ETDRS grade 43, or worse (microaneurysms plus minimal or mild cotton wool spots and intraretinal microvascular abnormalities, or worse); otherwise, B43 = 0. A variable for modified ETDRS grade 20, microaneurysms only, in worse eye at entry to study was tried but not found to be significant in the models described below. Blood samples for HbA1c were taken at study entry, at randomization, and annually thereafter. Biochemistry methods have been reported previously (UKPDS Group, 1994).
2.3. Statistical methods Parametric models for the risk for retinal photocoagulation are fitted. All models are described fully below. In overview, a reference model, in which risk depends on baseline retinopathy grade and duration of diabetes, is fitted; it is then generalized, to take account of the effect of glycemia measured by HbA1c, in two ways. In the first, HbA1c and duration of diabetes each have a proportional hazards effect on risk. In the second, HbA1 c and duration of diabetes are compounded into a single measure of glucose exposure over time. 2.4. Reference model For a patient who has not required photocoagulation t 1 years after diagnosis of diabetes, let P(t) denote the
Table 1 Characteristics of patients at entry to study, as mean, standard deviation or n (%) as indicated 3,648 patients 1,454 patients excluded in this analysis due to missing data Male 2,166 (59%) 842 (58%) White 3,050 (84%) 1,130 (78%) Afro-Caribbean 247 (7%) 140 (10%) Asian-Indian 325 (9%) 171 (12%) Other ethnic group 26 (0.7%) 13 (0.9%) Smoker 1,114 (31%) 464 (32%) Retinopathy grade 35 372 (10.2%) 8 (11.0%)* Retinopathy grade 43 or worse 127 (3.5%) 4 (5.5%)* Age (years) 52.3, 8.8 52.5, 8.8 HbA1C (%) 7.2, 1.8 7.2, 1.8y Systolic blood pressure (mm Hg) 135, 19 136, 20 27.7, 5.3 27.2, 5.4 Body mass index (kg/m2) * Data available for 73 patients. y Data available for 1,116 patients.
R.J. Stevens et al. / Journal of Diabetes and Its Complications 16 (2002) 371–376
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probability of requiring photocoagulation within the next year, that is, by time t. Prevalence of retinopathy increases with time since diagnosis of diabetes, rather than with age (UKPDS Group, 1997). Risk for retinal photocoagulation also increases with evidence of retinopathy at diagnosis of diabetes (UKPDS Group, 2000b). Hence, the model form
2.6. Glucose exposure model
B35 B43 t1 PðtÞ ¼ b0 r35 r43 d0
Again, the special case l = 0 is equivalent to Eq. (1), allowing a likelihood-ratio test for the value of the glucose exposure model. This model form also induces an approximately exponential relationship between P(t) and HbA1c. This is a model in which the effect of HbA1c is cumulative. In Eq. (1), risk increased by a constant ratio d0 each year, but in Eq. (3), the risk increase each year varies with the HbA1c measurement for that year. Total modeled risk for a given year is therefore dependent on the HbA1c values recorded in all preceding years.
ð1Þ
is proposed. Here r35 and r43 are risk ratios for the presence of the corresponding retinopathy states at study entry; d0 is a risk ratio for each year of duration of diabetes; and b0 is an intercept parameter, the value of P (Klein et al., 1995) for a patient with no retinopathy, or microaneurysms but no other retinopathy, at diagnosis of diabetes. To examine the effect of HbA1c on risk of photocoagulation, this paper considers two possible extensions to the above model. 2.5. Proportional hazards model In a proportional hazards model, P(t) varies directly with HbA1c. Since HbA1c measurements were taken annually in the UKPDS, it can be used as an updated covariate. If ht is the HbA1c measurement recorded at time t, the updated covariates model is B35 B43 t1 r43 d0 PðtÞ ¼ bðht Þr35
ð2Þ
where b(h) is some function of h. Observational analysis of the UKPDS has found an approximately exponential relationship between HbA1c and hazard for microvascular events (UKPDS Group, 2000a), motivating the form bðhÞ ¼ expða þ bðh 6:0ÞÞ: The special case b = 0 is identical to equation [1], which allows comparison of the two by a likelihood-ratio test. In this model, the effect of HbA1c on risk is immediate and short-lived, in the sense that only the most recent HbA1c value informs risk.
An alternative generalization of Eq. (1) allows h to act on the duration parameter d, as follows: B35 B43 r43 dðh1 Þdðh2 Þ . . . dðht1 Þ; PðtÞ ¼ b0 r35
where dðhÞ ¼ 1:0 þ lðh kÞ
ð3Þ
2.7. Model fitting and evaluation Model parameters were fitted by maximum likelihood estimation, and likelihood ratio methods were used to test for redundant or equivalent parameters (Bain & Engelhardt, 1991). Programming was carried out in the C language (Kernighan & Ritchie, 1988), with subroutines from the NAG Fortran library (Numerical Algorithms Group, 1995). To meet the probability theory requirement that probabilities do not exceed 1, the approximation Q(t) = 1 exp( P(t)) was used in fitting and in evaluation. For all typical values, P(t) is very closely approximated by Q(t), so that model interpretations are unaffected. For example, at P = .005, Q = 0.004988. The extended models (2) and (3) are compared to the reference model (1) in likelihood ratio tests. To compare the extended models to each other, we use graphical methods, since likelihood ratio tests are not valid for non-nested models. Modeled survival rates are plotted against survival rates observed in the study, together with 95% confidence intervals. The model predictions were calculated by simulation; averaging results for all 3648 patients over 50 iterations of the simulation program was found to provide
Table 2 Maximum likelihood estimates (95% confidence intervals) for parameters, with likelihood ratio comparisons to the duration model Model
b0, b(h)
r35
r43
d0, d(h)
Change in log-likelihood
P-value
Reference [1] Proportional Hazards [2]
b0 = 0.0030 (0.0023, 0.0037) b(h) = exp(a+b(h6.0)) where a = 5.9 (6.2, 5.7), b = 0.13 (0.09, 0.18) b0 = 0.0033 (0.0024, 0.0041)
5.0 (3.7, 6.3) 5.0 (3.8, 6.3)
23.4 (17.0, 29.7) 24.0 (17.5, 30.6)
d0 = 1.11 (1.08, 1.14) d0 = 1.09 (1.06, 1.12)
13.7
<106
5.1 (3.8, 6.4)
23.2 (16.9, 29.6)
d(h) = 1+l (h – k) where k = 6.0 (5.1, 6.8), l = 0.046 (0.038, 0.055)
53.6
<106
Glucose exposure [3]
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three models. Both models (2) and (3) show highly significant improvement on model (1), confirming the importance of HbA1c as a risk factor for retinal photocoagulation. 3.2. Predictive power Fig. 1a shows event rates observed in the study for the 3648 patients, calculated by a life-table method, together with 95% confidence limits, superimposed on the corresponding model predictions according to model (2). Fig. 1b shows the predictions from model (3) to be an equally good match for the observed data. Of the 3648 patients in this analysis, 2975 took part in the UKPDS blood glucose trial, being randomised either to an intensive or to a conventional blood glucose control policy. In Fig. 2, the modeled and observed event rates for these patients are shown separately for the conventional and intensive groups. Fig. 2a indicates that, according to the proportional hazards model (2), event rates for photocoagulation should have been quite similar in the two arms of the study; this model fails to capture the effect of blood glucose lowering on risk of photocoagulation. In Fig. 2b, predictions from the glucose exposure model (3) are seen to be consid-
Fig. 1. Observed survival rates (+), with 95% confidence limits (dashed lines), by life-tables method, compared to modeled survival rates (solid line) from (a) model 2 and (b) model 3.
stable results. Within-study event rates, and 95% confidence intervals, were calculated by a life-table method. In a sensitivity analysis to evaluate the importance of the regression dilution effect, all models were also fitted to a smoothed data set, generated as follows. For each patient, a linear regression line was fitted to all available HbA1c data from year 1 onwards. Then missing HbA1c values for the patient were replaced by estimates from the regression line, as in the main analysis, but the original data values were also replaced by regression line estimates. The models were then fitted to this smoothed data set, in which each data point has as much stability against measurement error as possible.
3. Results During follow-up, 303 events occurred in 3648 patients. Median time to event or censoring was 11 years. 3.1. Fitted values Table 2 shows fitted values and the corresponding loglikelihoods for each of the three model equations. Parameter estimates, for example, for r35 and r43 are stable across all
Fig. 2. Observed survival rates, by life-tables method, by UKPDS therapy group (O conventional group, intensive group), compared to modeled survival rates (solid line for conventional, dashed line for intensive group) by (a) model 2 and (b) model 3.
R.J. Stevens et al. / Journal of Diabetes and Its Complications 16 (2002) 371–376
erably closer to the observed event rates. Hence model (3) succeeds in explaining most of the treatment effect observed in the UKPDS as a consequence of HbA1c lowering. 3.3. Regression dilution Since the proportional hazards model (2) uses only a single value of HbA1c, which is subject to measurement error, at each time point, we considered the possibility that its weaker predictive power was due to a regression dilution effect (Frost & Thompson, 2000). To test for this, all analyses were repeated with a smoothed data set in which HbA1c was smoothed by linear regression. Again, both models (2) and (3) were significantly better than reference model (1), and the predicted survival curves generated by each model were not visibly different from those in Figs. 1 and 2.
4. Discussion Previous studies have shown that the risk for retinal photocoagulation varies continuously with glycemia (Klein et al., 1995; Nakagami et al., 1997), and increases with duration of diabetes (UKPDS Group, 1997). These results are confirmed by the likelihood ratio tests, in which both models (2) and (3) are significant improvements on reference model (1). New here are direct comparisons between models (2) and (3), in the graphical results. Fig. 1 indicates that model (3) fits the data at least as well as model (2): thus, duration is not necessarily an independent risk factor for photocoagulation, but may alternatively be only one component of a single risk factor, cumulative glucose exposure over time. Fig. 2 suggests further that the glucose exposure model is superior, better reflecting the differences between a high-HbA1c and a low-HbA1c cohort. The models presented here should be regarded as a mathematical framework for a proof of concept, rather than generalizable risk equations, which would require consideration of risk factors for development and progression of retinopathy such as sex and blood pressure (Klein, Klein, Moss, Davis, & DeMets, 1989; UKPDS Group, 2000b). Our endpoint does not distinguish between pan-retinal photocoagulation, and focal photocoagulation, but this does not affect our use of the endpoint as a marker of microvascular damage, as both proliferative retinopathy and macular edema are microvascular diseases of the eye. The variable t, duration of diagnosed diabetes, has been used throughout as a surrogate for the unknown actual duration of diabetes; this will diminish accuracy in each model, although the adjustment for baseline retinopathy will help to correct for this. That the levels of retinopathy reported in Table 1 are lower than reported in the Wisconsin Epidemiologic Study of Diabetic Retinopathy is likely to be because the UKPDS recruited patients with newly diagnosed diabetes, whereas those in WESDR had a mean 14-year duration of diagnosed diabetes (Klein, Klein, Moss, & Cruickshanks, 1994).
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The success of the glucose exposure model (3) further implies that, in diabetic patients with particularly good glycemic control, hazard would not increase over time but hold steady or decrease. The threshold at which this occurs is indicated by the estimated intercept parameter k, which is HbA1c = 6.0%. It would be desirable to verify that this is not an artefact of a linear model for d(h), for example, by analysis of a subgroup with HbA1c consistently lower than 6.0%. This cohort includes too few patients in whom HbA1c does not exceed 6.0% after study entry (10 cases of photocoagulation among 179 patients), but such an effect has been observed in type 1 diabetes in the DCCT, as discussed above (DCCT Research Group, 1995). This paper has shown that two well-known risk factors for retinal photocoagulation are better considered as a single, cumulative exposure variable, offering researchers a simpler model that also has greater predictive power. The models also have clinical implications: in type 2 diabetes as in type 1, the increasing risk for photocoagulation with time is a consequence of elevated glycemia, and hence potentially avoidable.
Acknowledgments The major grants for this study were from The National Eye Institute and The National Institute of Digestive, Diabetes and Kidney Disease in the National Institutes of Health, USA (grant 3 U10 EY07049), the UK Medical Research Council, British Diabetic Association, the UK Department of Health, The British Heart Foundation, NovoNordisk, Bayer, Bristol Myers Squibb, Hoechst, Lilly, Lipha and Farmitalia Carlo Erba. R.S. was supported by a Wellcome Trust fellowship (grant 054470/Z/98/Z/DG/NOS/FH). The authors are grateful to Steve Aldington for advice on opthalmic aspects of this paper.
References Bain, L. J., & Engelhardt, M. (1991). Introduction to probability and mathematical statistics (2nd ed.). Boston: PWS-Kent (Sciences TDASiSaD, ed.). Cull, C. A., Manley, S. E., Stratton, I. M., Neil, H. A. W., Ross, I. S., Holman, R. R., Turner, R. C., & Matthews, D. R. (1997). Approach to maintaining comparability of biochemical data during long-term clinical trials. Clinical Chemistry, 43, 1913 – 1918. DCCT Research Group. (1995). The relationship of glycemic exposure (HbA1c) to the risk of development and progression of retinopathy in the Diabetes Control and Complications Trial. Diabetes, 44, 968 – 983. Dowse, G. K., Humphrey, A. R. G., Collins, V. R., Plehwe, W., Gareedo, H., Fareed, D., Hemraj, F., Taylor, H. R., Tuomilehto, J., Alberti, K.G.M.M., & Zimmet, P. Z. (1998). Prevalence and risk factors for diabetic retinopathy in the multi-ethnic population of Mauritius. American Journal of Epidemiology, 147, 448 – 457. ETDRS Research Group. (1991). Fundus photographic risk factors for progression of diabetic retinopathy: ETDRS 12. Ophthalmology, 98, 823 – 833.
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Frost, C., & Thompson, S. (2000). Correcting for regression dilution bias: comparison of methods for a single predictor variable. Journal of the Royal Statistical Series A, 163, 173 – 190. Kernighan, B. W., & Ritchie, D. M. (1988). In: B. W. Kernighan (Ed.), The C programming language (2nd ed.). New Jersey: Prentice-Hall (Prentice Hall Software Series). Klein, R., Klein, B. E., & Moss, S. E. (1995). Epidemiology of proliferative diabetic retinopathy. Diabetes Care, 15, 1875 – 1891. Klein, R., Klein, B. E. K., Moss, S. E., & Cruickshanks, K. J. (1994). WESDR XIV. Ten-year incidence and progression of diabetic retinopathy. Archives of Ophthalmology, 112, 1217 – 1228. Klein, R., Klein, B. E. K., Moss, S. E., Davis, M. D., & DeMets, D. L. (1989). Is blood pressure a predictor of the incidence for progression of diabetic retinopathy? Archives of Internal Medicine, 149, 2427 – 2432. Nakagami, T., Kawahara, R., Hori, S., & Omori, Y. (1997). Glycemic control and prevention of retinopathy in Japanese NIDDM patients. Diabetes Care, 20, 621 – 623. Numerical Algorithms Group. (1995). NAG Fortran library manual (17th ed.). Oxford: Numerical Algorithms Group. Ohkubo, Y., Kishikawa, H., Araki, E., Miyata, T., Isami, S., Motoyoshi, S., Kojima, Y., Furuyoshi, N., & Shichiri, M. (1995). Intensive insulin therapy prevents the progression of diabetic microvascular complications in Japanese patients with non-insulin-dependent diabetes mellitus: a randomized prospective 6-year study. Diabetes Research and Clinical Practice, 28, 103 – 117. Shorr, R. I., Franse, L. V., Resnick, H. E., Bari, M. D., Johnson, K. C., & Pahor, M. (2000). Glycemic control of older adults with type 2 diabetes: findings from the third national health and nutrition examination survey, 1988 – 1994. Journal of the American Geriatrics Society, 48, 264 – 267.
UKPDS Group. (1991). UK Prospective Diabetes Study VIII: study design, progress and performance. Diabetologia, 34, 877 – 890. UKPDS Group. (1994). UK Prospective Diabetes Study XI: biochemical risk factors in type 2 diabetic patients at diagnosis compared with age-matched normal subjects. Diabetic Medicine, 11, 534 – 544. UKPDS Group. (1996). UK Prospective Diabetes Study 17: a nine-year update of a randomized, controlled trial on the effect of improved metabolic control on complications in non-insulin-dependent diabetes mellitus. Annals of Internal Medicine, 124, 136 – 145. UKPDS Group. (1997). UK Prospective Diabetes Study 22: effect of age at diagnosis on diabetic tissue damage during the first 6 years of NIDDM. Diabetes Care, 20, 1435 – 1441. UKPDS Group. (1998a). Intensive blood glucose control with sulphonylureas or insulin compared with conventional treatment and risk of complications in patients with type 2 diabetes (UKPDS 33). Lancet, 352, 837 – 853. UKPDS Group. (1998b). UK Prospective Diabetes Study 30: diabetic retinopathy at diagnosis of type 2 diabetes and associated risk factors. Archives of Ophthalmology, 116, 297 – 303. UKPDS Group. (1999). Glycaemic control with diet, sulphonylurea, metformin and insulin therapy in patients with type 2 diabetes: progressive requirement for multiple therapies (UKPDS 49). Journal of the American Medical Association, 281, 2005 – 2012. UKPDS Group. (2000a). Association of glycaemia with macrovascular and microvascular complications of type 2 diabetes (UKPDS 35). British Medical Journal, 321, 405 – 412. UKPDS Group. (2000b). UKPDS 50: Risk factors for incidence and progression of retinopathy in type 2 diabetes over six years from diagnosis. Diabetologia, 44, 156 – 163.
UKPDS58—modeling glucose exposure as a risk factor for photocoagulation in type 2 diabetes Richard J. Stevens*, Irene M. Stratton, Rury R. Holman Diabetes Trials Unit, Oxford Centre for Diabetes, Endocrinology and Metabolism, University of Oxford, Oxford, UK Received 4 October 2001; received in revised form 18 December 2001; accepted 28 December 2001
Abstract In type 2 diabetes, the risk of retinopathy, and of retinal photocoagulation, rises with time after diagnosis of diabetes. In this paper, mathematical modeling shows that this ageing effect is attributable to the rise in glycemia with time since diagnosis of diabetes. Mathematical models were fitted to data from 3648 patients from the UK Prospective Diabetes Study (UKPDS). A proportional hazards model, in which time and glycemia measured by HbA1c are independent risk factors for photocoagulation, was compared to a model in which time does not contribute except through a measure of cumulative glucose exposure. Since likelihood ratio tests cannot be applied to non-nested models, graphical methods were used to compare the two models. The glucose exposure model was able to fit variation in survival with time at least as well as the proportional hazards model. The proportional hazards model, however, seriously underestimates the differences in two groups of different mean HbA1c. We conclude that duration of diabetes and HbA1c level better predict risk for photocoagulation when treated as two components of cumulative glucose exposure, than when treated as independent risk factors. D 2002 Elsevier Science Inc. All rights reserved. Keywords: Modeling; Glucose exposure; Duration of diabetes; Age; Retinal photocoagulation
1. Introduction Glycated hemoglobin, and its subfraction HbA1c, provide integrated measures of the degree of hyperglycemia in an individual over the preceding 2 –3 months. Previous studies have found that an elevated HbA1c or glycated hemoglobin indicates an increased risk of diabetic retinopathy (Klein, Klein, & Moss, 1995; Nakagami, Kawahara, Hori, & Omori, 1997). Randomized controlled clinical trials in both type 1 and type 2 diabetic subjects have demonstrated that improved glycemic control can reduce risk of progression of retinopathy, suggesting a causative relationship between glycemia and retinopathy (DCCT Research Group, 1995; Ohkubo et al., 1995; UKPDS Group, 1998a). It is also known that the risk for retinal photocoagulation rises with duration of diabetes in both type 1 and type 2 (Klein et al., 1995; UKPDS Group, 1997), and that HbA1c levels tend to rise with duration of diabetes (Shorr et al., 2000; UKPDS Group, 1999). In type 2 diabetes, duration of * Corresponding author. Diabetes Trials Unit, Radcliffe Infirmary, 2nd Floor, Woodstock Road, Block 11, Oxford OX2 6HE, UK. Tel.: +44-1865224463; fax: +44-1865-723884. E-mail address: [email protected] (R.J. Stevens).
diabetes is a risk factor for retinopathy, but age at diagnosis of diabetes is not (UKPDS Group, 1997). Thus, in type 2 diabetes, age is a risk factor only as a marker for duration of diabetes. Given that both glycemia and duration of diabetes are risk factors for retinopathy and for retinal photocoagulation, one possibility is that these effects are independent. Another is that the increased risk over time is attributable in whole or large part to cumulative exposure to high levels of HbA1c over time (Dowse et al., 1998). In type 1 diabetes, the latter hypothesis is supported by evidence from The Diabetes Control and Complications Trials (DCCT), in which patients with the lowest HbA1c levels had no increase in risk for retinopathy progression over time (DCCT Research Group, 1995). The present paper explores the same hypothesis for type 2 diabetic patients, using data from the UK Prospective Diabetes Study (UKPDS), through the use of statistical models. The relationship between HbA1c and risk for retinal photocoagulation has been reported in a previous UKPDS paper (UKPDS Group, 2000a). The aim of the present paper is to use statistical models to explore two alternative hypotheses, not to present a fully generalizable risk model. In the first hypothesis, duration of diagnosed
1056-8727/02/$ – see front matter D 2002 Elsevier Science Inc. All rights reserved. PII: S 1 0 5 6 - 8 7 2 7 ( 0 2 ) 0 0 1 6 1 - 7
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R.J. Stevens et al. / Journal of Diabetes and Its Complications 16 (2002) 371–376
diabetes, and recent HbA1c, are risk factors that act independently and multiplicatively, as in a proportional hazards model. In the second, cumulative exposure to high levels of HbA1c acts directly on risk for photocoagulation, and duration of diabetes does not affect risk directly but through increased cumulative HbA1c exposure. Each of these can be represented by a parametric model, and the fit of each model to the observed data can be evaluated.
2. Methods
HbA1c was measured by DCCT-aligned (UKPDS Group, 1998a) HPLC (Biorad Diamat Automated Glycosylated Haemoglobin Analyser), non-diabetic range 4.5– 6.2% (Cull et al., 1997; UKPDS Group, 1994). There were 3648 patients, with both baseline retinopathy grade and follow-up HbA1c data, included in this analysis. Patients with photocoagulation prior to diagnosis of diabetes were excluded by the UKPDS protocol. Table 1 shows baseline characteristics of the patients. For patients with incomplete HbA1c data, linear regression was used to interpolate and extrapolate missing values. This was necessary for 1768 patients, for a total of 4448 values.
2.1. Patients
2.2. Endpoint
The UKPDS recruited 5102 patients with newly diagnosed type 2 diabetes, in 23 centres, between 1977 and 1991, and continued until the study closed on 30th September 1997. In addition to clinician diagnosis of diabetes, patients were included in the study if their fasting plasma glucose exceeded 6 mmol/L on two further occasions. Exclusion criteria were severe vascular disease, accelerated hypertension, severe retinopathy requiring photocoagulation, renal failure (serume creatinine > 175 mmol/L), life threatening disease, disease requiring steroid treatment, or occupation precluding treatment with insulin. The UKPDS protocol has been described in detail elsewhere (UKPDS Group, 1991, 1996). The study design conformed to the Declarations of Helsinki (1975 and 1983), and was approved by the local Research Ethics Committee in each centre. Each patient gave informed, witnessed consent. Patients in 22 centres recruited since 1983 had retinopathy assessed by four-field retinal photography on study entry and every 3 years thereafter. At each assessment, a grade was assigned according to a modification of the ETDRS scale (ETDRS Research Group, 1991). Details of the photographic methods have been described elsewhere (UKPDS Group, 1998b). For this analysis, we define indicator variables B35 and B43 for modified retinopathy grade in the worse eye at study entry, as follows:
The decision for photocoagulation was taken by opthalmologists independent of the UKPDS (UKPDS Group, 1998a). We include both pan-retinal photocoagulation, and focal photocoagulation for macular edema.
define B35 = 1 for modified ETDRS grade 35 (microaneurysms plus minimal haemorrhages and/or hard exudates), otherwise, B35 = 0; and B43 = 1 for modified ETDRS grade 43, or worse (microaneurysms plus minimal or mild cotton wool spots and intraretinal microvascular abnormalities, or worse); otherwise, B43 = 0. A variable for modified ETDRS grade 20, microaneurysms only, in worse eye at entry to study was tried but not found to be significant in the models described below. Blood samples for HbA1c were taken at study entry, at randomization, and annually thereafter. Biochemistry methods have been reported previously (UKPDS Group, 1994).
2.3. Statistical methods Parametric models for the risk for retinal photocoagulation are fitted. All models are described fully below. In overview, a reference model, in which risk depends on baseline retinopathy grade and duration of diabetes, is fitted; it is then generalized, to take account of the effect of glycemia measured by HbA1c, in two ways. In the first, HbA1c and duration of diabetes each have a proportional hazards effect on risk. In the second, HbA1 c and duration of diabetes are compounded into a single measure of glucose exposure over time. 2.4. Reference model For a patient who has not required photocoagulation t 1 years after diagnosis of diabetes, let P(t) denote the
Table 1 Characteristics of patients at entry to study, as mean, standard deviation or n (%) as indicated 3,648 patients 1,454 patients excluded in this analysis due to missing data Male 2,166 (59%) 842 (58%) White 3,050 (84%) 1,130 (78%) Afro-Caribbean 247 (7%) 140 (10%) Asian-Indian 325 (9%) 171 (12%) Other ethnic group 26 (0.7%) 13 (0.9%) Smoker 1,114 (31%) 464 (32%) Retinopathy grade 35 372 (10.2%) 8 (11.0%)* Retinopathy grade 43 or worse 127 (3.5%) 4 (5.5%)* Age (years) 52.3, 8.8 52.5, 8.8 HbA1C (%) 7.2, 1.8 7.2, 1.8y Systolic blood pressure (mm Hg) 135, 19 136, 20 27.7, 5.3 27.2, 5.4 Body mass index (kg/m2) * Data available for 73 patients. y Data available for 1,116 patients.
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probability of requiring photocoagulation within the next year, that is, by time t. Prevalence of retinopathy increases with time since diagnosis of diabetes, rather than with age (UKPDS Group, 1997). Risk for retinal photocoagulation also increases with evidence of retinopathy at diagnosis of diabetes (UKPDS Group, 2000b). Hence, the model form
2.6. Glucose exposure model
B35 B43 t1 PðtÞ ¼ b0 r35 r43 d0
Again, the special case l = 0 is equivalent to Eq. (1), allowing a likelihood-ratio test for the value of the glucose exposure model. This model form also induces an approximately exponential relationship between P(t) and HbA1c. This is a model in which the effect of HbA1c is cumulative. In Eq. (1), risk increased by a constant ratio d0 each year, but in Eq. (3), the risk increase each year varies with the HbA1c measurement for that year. Total modeled risk for a given year is therefore dependent on the HbA1c values recorded in all preceding years.
ð1Þ
is proposed. Here r35 and r43 are risk ratios for the presence of the corresponding retinopathy states at study entry; d0 is a risk ratio for each year of duration of diabetes; and b0 is an intercept parameter, the value of P (Klein et al., 1995) for a patient with no retinopathy, or microaneurysms but no other retinopathy, at diagnosis of diabetes. To examine the effect of HbA1c on risk of photocoagulation, this paper considers two possible extensions to the above model. 2.5. Proportional hazards model In a proportional hazards model, P(t) varies directly with HbA1c. Since HbA1c measurements were taken annually in the UKPDS, it can be used as an updated covariate. If ht is the HbA1c measurement recorded at time t, the updated covariates model is B35 B43 t1 r43 d0 PðtÞ ¼ bðht Þr35
ð2Þ
where b(h) is some function of h. Observational analysis of the UKPDS has found an approximately exponential relationship between HbA1c and hazard for microvascular events (UKPDS Group, 2000a), motivating the form bðhÞ ¼ expða þ bðh 6:0ÞÞ: The special case b = 0 is identical to equation [1], which allows comparison of the two by a likelihood-ratio test. In this model, the effect of HbA1c on risk is immediate and short-lived, in the sense that only the most recent HbA1c value informs risk.
An alternative generalization of Eq. (1) allows h to act on the duration parameter d, as follows: B35 B43 r43 dðh1 Þdðh2 Þ . . . dðht1 Þ; PðtÞ ¼ b0 r35
where dðhÞ ¼ 1:0 þ lðh kÞ
ð3Þ
2.7. Model fitting and evaluation Model parameters were fitted by maximum likelihood estimation, and likelihood ratio methods were used to test for redundant or equivalent parameters (Bain & Engelhardt, 1991). Programming was carried out in the C language (Kernighan & Ritchie, 1988), with subroutines from the NAG Fortran library (Numerical Algorithms Group, 1995). To meet the probability theory requirement that probabilities do not exceed 1, the approximation Q(t) = 1 exp( P(t)) was used in fitting and in evaluation. For all typical values, P(t) is very closely approximated by Q(t), so that model interpretations are unaffected. For example, at P = .005, Q = 0.004988. The extended models (2) and (3) are compared to the reference model (1) in likelihood ratio tests. To compare the extended models to each other, we use graphical methods, since likelihood ratio tests are not valid for non-nested models. Modeled survival rates are plotted against survival rates observed in the study, together with 95% confidence intervals. The model predictions were calculated by simulation; averaging results for all 3648 patients over 50 iterations of the simulation program was found to provide
Table 2 Maximum likelihood estimates (95% confidence intervals) for parameters, with likelihood ratio comparisons to the duration model Model
b0, b(h)
r35
r43
d0, d(h)
Change in log-likelihood
P-value
Reference [1] Proportional Hazards [2]
b0 = 0.0030 (0.0023, 0.0037) b(h) = exp(a+b(h6.0)) where a = 5.9 (6.2, 5.7), b = 0.13 (0.09, 0.18) b0 = 0.0033 (0.0024, 0.0041)
5.0 (3.7, 6.3) 5.0 (3.8, 6.3)
23.4 (17.0, 29.7) 24.0 (17.5, 30.6)
d0 = 1.11 (1.08, 1.14) d0 = 1.09 (1.06, 1.12)
13.7
<106
5.1 (3.8, 6.4)
23.2 (16.9, 29.6)
d(h) = 1+l (h – k) where k = 6.0 (5.1, 6.8), l = 0.046 (0.038, 0.055)
53.6
<106
Glucose exposure [3]
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three models. Both models (2) and (3) show highly significant improvement on model (1), confirming the importance of HbA1c as a risk factor for retinal photocoagulation. 3.2. Predictive power Fig. 1a shows event rates observed in the study for the 3648 patients, calculated by a life-table method, together with 95% confidence limits, superimposed on the corresponding model predictions according to model (2). Fig. 1b shows the predictions from model (3) to be an equally good match for the observed data. Of the 3648 patients in this analysis, 2975 took part in the UKPDS blood glucose trial, being randomised either to an intensive or to a conventional blood glucose control policy. In Fig. 2, the modeled and observed event rates for these patients are shown separately for the conventional and intensive groups. Fig. 2a indicates that, according to the proportional hazards model (2), event rates for photocoagulation should have been quite similar in the two arms of the study; this model fails to capture the effect of blood glucose lowering on risk of photocoagulation. In Fig. 2b, predictions from the glucose exposure model (3) are seen to be consid-
Fig. 1. Observed survival rates (+), with 95% confidence limits (dashed lines), by life-tables method, compared to modeled survival rates (solid line) from (a) model 2 and (b) model 3.
stable results. Within-study event rates, and 95% confidence intervals, were calculated by a life-table method. In a sensitivity analysis to evaluate the importance of the regression dilution effect, all models were also fitted to a smoothed data set, generated as follows. For each patient, a linear regression line was fitted to all available HbA1c data from year 1 onwards. Then missing HbA1c values for the patient were replaced by estimates from the regression line, as in the main analysis, but the original data values were also replaced by regression line estimates. The models were then fitted to this smoothed data set, in which each data point has as much stability against measurement error as possible.
3. Results During follow-up, 303 events occurred in 3648 patients. Median time to event or censoring was 11 years. 3.1. Fitted values Table 2 shows fitted values and the corresponding loglikelihoods for each of the three model equations. Parameter estimates, for example, for r35 and r43 are stable across all
Fig. 2. Observed survival rates, by life-tables method, by UKPDS therapy group (O conventional group, intensive group), compared to modeled survival rates (solid line for conventional, dashed line for intensive group) by (a) model 2 and (b) model 3.
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erably closer to the observed event rates. Hence model (3) succeeds in explaining most of the treatment effect observed in the UKPDS as a consequence of HbA1c lowering. 3.3. Regression dilution Since the proportional hazards model (2) uses only a single value of HbA1c, which is subject to measurement error, at each time point, we considered the possibility that its weaker predictive power was due to a regression dilution effect (Frost & Thompson, 2000). To test for this, all analyses were repeated with a smoothed data set in which HbA1c was smoothed by linear regression. Again, both models (2) and (3) were significantly better than reference model (1), and the predicted survival curves generated by each model were not visibly different from those in Figs. 1 and 2.
4. Discussion Previous studies have shown that the risk for retinal photocoagulation varies continuously with glycemia (Klein et al., 1995; Nakagami et al., 1997), and increases with duration of diabetes (UKPDS Group, 1997). These results are confirmed by the likelihood ratio tests, in which both models (2) and (3) are significant improvements on reference model (1). New here are direct comparisons between models (2) and (3), in the graphical results. Fig. 1 indicates that model (3) fits the data at least as well as model (2): thus, duration is not necessarily an independent risk factor for photocoagulation, but may alternatively be only one component of a single risk factor, cumulative glucose exposure over time. Fig. 2 suggests further that the glucose exposure model is superior, better reflecting the differences between a high-HbA1c and a low-HbA1c cohort. The models presented here should be regarded as a mathematical framework for a proof of concept, rather than generalizable risk equations, which would require consideration of risk factors for development and progression of retinopathy such as sex and blood pressure (Klein, Klein, Moss, Davis, & DeMets, 1989; UKPDS Group, 2000b). Our endpoint does not distinguish between pan-retinal photocoagulation, and focal photocoagulation, but this does not affect our use of the endpoint as a marker of microvascular damage, as both proliferative retinopathy and macular edema are microvascular diseases of the eye. The variable t, duration of diagnosed diabetes, has been used throughout as a surrogate for the unknown actual duration of diabetes; this will diminish accuracy in each model, although the adjustment for baseline retinopathy will help to correct for this. That the levels of retinopathy reported in Table 1 are lower than reported in the Wisconsin Epidemiologic Study of Diabetic Retinopathy is likely to be because the UKPDS recruited patients with newly diagnosed diabetes, whereas those in WESDR had a mean 14-year duration of diagnosed diabetes (Klein, Klein, Moss, & Cruickshanks, 1994).
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The success of the glucose exposure model (3) further implies that, in diabetic patients with particularly good glycemic control, hazard would not increase over time but hold steady or decrease. The threshold at which this occurs is indicated by the estimated intercept parameter k, which is HbA1c = 6.0%. It would be desirable to verify that this is not an artefact of a linear model for d(h), for example, by analysis of a subgroup with HbA1c consistently lower than 6.0%. This cohort includes too few patients in whom HbA1c does not exceed 6.0% after study entry (10 cases of photocoagulation among 179 patients), but such an effect has been observed in type 1 diabetes in the DCCT, as discussed above (DCCT Research Group, 1995). This paper has shown that two well-known risk factors for retinal photocoagulation are better considered as a single, cumulative exposure variable, offering researchers a simpler model that also has greater predictive power. The models also have clinical implications: in type 2 diabetes as in type 1, the increasing risk for photocoagulation with time is a consequence of elevated glycemia, and hence potentially avoidable.
Acknowledgments The major grants for this study were from The National Eye Institute and The National Institute of Digestive, Diabetes and Kidney Disease in the National Institutes of Health, USA (grant 3 U10 EY07049), the UK Medical Research Council, British Diabetic Association, the UK Department of Health, The British Heart Foundation, NovoNordisk, Bayer, Bristol Myers Squibb, Hoechst, Lilly, Lipha and Farmitalia Carlo Erba. R.S. was supported by a Wellcome Trust fellowship (grant 054470/Z/98/Z/DG/NOS/FH). The authors are grateful to Steve Aldington for advice on opthalmic aspects of this paper.
References Bain, L. J., & Engelhardt, M. (1991). Introduction to probability and mathematical statistics (2nd ed.). Boston: PWS-Kent (Sciences TDASiSaD, ed.). Cull, C. A., Manley, S. E., Stratton, I. M., Neil, H. A. W., Ross, I. S., Holman, R. R., Turner, R. C., & Matthews, D. R. (1997). Approach to maintaining comparability of biochemical data during long-term clinical trials. Clinical Chemistry, 43, 1913 – 1918. DCCT Research Group. (1995). The relationship of glycemic exposure (HbA1c) to the risk of development and progression of retinopathy in the Diabetes Control and Complications Trial. Diabetes, 44, 968 – 983. Dowse, G. K., Humphrey, A. R. G., Collins, V. R., Plehwe, W., Gareedo, H., Fareed, D., Hemraj, F., Taylor, H. R., Tuomilehto, J., Alberti, K.G.M.M., & Zimmet, P. Z. (1998). Prevalence and risk factors for diabetic retinopathy in the multi-ethnic population of Mauritius. American Journal of Epidemiology, 147, 448 – 457. ETDRS Research Group. (1991). Fundus photographic risk factors for progression of diabetic retinopathy: ETDRS 12. Ophthalmology, 98, 823 – 833.
376
R.J. Stevens et al. / Journal of Diabetes and Its Complications 16 (2002) 371–376
Frost, C., & Thompson, S. (2000). Correcting for regression dilution bias: comparison of methods for a single predictor variable. Journal of the Royal Statistical Series A, 163, 173 – 190. Kernighan, B. W., & Ritchie, D. M. (1988). In: B. W. Kernighan (Ed.), The C programming language (2nd ed.). New Jersey: Prentice-Hall (Prentice Hall Software Series). Klein, R., Klein, B. E., & Moss, S. E. (1995). Epidemiology of proliferative diabetic retinopathy. Diabetes Care, 15, 1875 – 1891. Klein, R., Klein, B. E. K., Moss, S. E., & Cruickshanks, K. J. (1994). WESDR XIV. Ten-year incidence and progression of diabetic retinopathy. Archives of Ophthalmology, 112, 1217 – 1228. Klein, R., Klein, B. E. K., Moss, S. E., Davis, M. D., & DeMets, D. L. (1989). Is blood pressure a predictor of the incidence for progression of diabetic retinopathy? Archives of Internal Medicine, 149, 2427 – 2432. Nakagami, T., Kawahara, R., Hori, S., & Omori, Y. (1997). Glycemic control and prevention of retinopathy in Japanese NIDDM patients. Diabetes Care, 20, 621 – 623. Numerical Algorithms Group. (1995). NAG Fortran library manual (17th ed.). Oxford: Numerical Algorithms Group. Ohkubo, Y., Kishikawa, H., Araki, E., Miyata, T., Isami, S., Motoyoshi, S., Kojima, Y., Furuyoshi, N., & Shichiri, M. (1995). Intensive insulin therapy prevents the progression of diabetic microvascular complications in Japanese patients with non-insulin-dependent diabetes mellitus: a randomized prospective 6-year study. Diabetes Research and Clinical Practice, 28, 103 – 117. Shorr, R. I., Franse, L. V., Resnick, H. E., Bari, M. D., Johnson, K. C., & Pahor, M. (2000). Glycemic control of older adults with type 2 diabetes: findings from the third national health and nutrition examination survey, 1988 – 1994. Journal of the American Geriatrics Society, 48, 264 – 267.
UKPDS Group. (1991). UK Prospective Diabetes Study VIII: study design, progress and performance. Diabetologia, 34, 877 – 890. UKPDS Group. (1994). UK Prospective Diabetes Study XI: biochemical risk factors in type 2 diabetic patients at diagnosis compared with age-matched normal subjects. Diabetic Medicine, 11, 534 – 544. UKPDS Group. (1996). UK Prospective Diabetes Study 17: a nine-year update of a randomized, controlled trial on the effect of improved metabolic control on complications in non-insulin-dependent diabetes mellitus. Annals of Internal Medicine, 124, 136 – 145. UKPDS Group. (1997). UK Prospective Diabetes Study 22: effect of age at diagnosis on diabetic tissue damage during the first 6 years of NIDDM. Diabetes Care, 20, 1435 – 1441. UKPDS Group. (1998a). Intensive blood glucose control with sulphonylureas or insulin compared with conventional treatment and risk of complications in patients with type 2 diabetes (UKPDS 33). Lancet, 352, 837 – 853. UKPDS Group. (1998b). UK Prospective Diabetes Study 30: diabetic retinopathy at diagnosis of type 2 diabetes and associated risk factors. Archives of Ophthalmology, 116, 297 – 303. UKPDS Group. (1999). Glycaemic control with diet, sulphonylurea, metformin and insulin therapy in patients with type 2 diabetes: progressive requirement for multiple therapies (UKPDS 49). Journal of the American Medical Association, 281, 2005 – 2012. UKPDS Group. (2000a). Association of glycaemia with macrovascular and microvascular complications of type 2 diabetes (UKPDS 35). British Medical Journal, 321, 405 – 412. UKPDS Group. (2000b). UKPDS 50: Risk factors for incidence and progression of retinopathy in type 2 diabetes over six years from diagnosis. Diabetologia, 44, 156 – 163.