Stroke-free duration and stroke risk in patients with atrial fibrillation: simulation using a Bayesian inference

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1 Asian Biomedicine Vol. 3 No. 4 August 2009; Brief Communication (Original) Stroke-free duration and stroke risk in patients with atrial fibrillation: simulation using a Bayesian inference Tomoki Nakamizo, Masahiro Yamamoto Department of Neurology, Yokohama Stroke and Brain Center, Yokohama , Japan. Background: Prior stroke or transient ischemic attack (TIA) is a major predictor of stroke in patients with atrial fibrillation (AF). It may be regarded as a Bayesian predictor by which some underlying causes can be inferred from the result. Similarly, stroke-free duration may be a predictor of freedom from stroke, but this subject has remained unaddressed. Objective: To evaluate the above hypothesis by simulating how stroke-free duration can affect estimated stroke risk in patients with AF using a Bayesian inference. Methods: The development of stroke was modeled as a yearly Bernoulli trial. A Bayesian inference method was applied to the model with correction for the risk increase by aging. The model was composed of two risk parameters: prior stroke (or TIA) and aging. The parameters were treated as random variables. Their distributions were determined to agree with the clinical data reported in the meta-analysis by Stroke Risk in Atrial Fibrillation Working Group. In the simulation, we noted the stroke risk after 1,..,10 stroke-free years for initial risk 1,..,8 %/year. Results: Increasing stroke-free duration reduced the estimated stroke risk, offsetting the risk increase by aging. In patients whose stroke (including TIA) risk was 3%/year or less, the estimated risk remained almost constant with aging if they were stroke-free. In patients whose risk was 4%/year or higher, the risk decreased if they were stroke-free. Conclusion: Our analysis predicted that stroke-free duration in patients with AF is an important predictor of freedom from stroke. Increasing stroke-free duration will reduce the estimated stroke-risk, which may influence our decision to start anticoagulation. Stroke-free duration should be considered when estimating stroke risk in patients with AF. Keywords: Atrial fibrillation, Bayesian inference method, cardioembolism, clinical prediction, disease-free duration, probabilistic model, risk stratification, stroke. Atrial fibrillation (AF) is an important risk factor for stroke. In patients with AF, we must estimate stroke risk before starting anticoagulation, because it decreases stroke risk but increases the risk of bleeding complication. Predictors of stroke have been identified to estimate stroke risk. Among several predictors of stroke, a major predictor is prior stroke or transient ischemic attack (TIA) [1-3]. This predictor is different from other predictors, such as age, hypertension, diabetes, and Correspondence to: Tomoki Nakamizo, MD, Department of Neurology, Yokohama Stroke and Brain Center, Takigashira, Isogoku, Yokohama , Japan. to00- nakamizo@city.yokohama.jp congestive heart failure. It is reasonable to suppose a causal relationship between these factors and stroke. For example, heart failure promotes stasis in the left atrium, which leads to thrombosis. In contrast, when prior stroke or TIA is used as a predictor of stroke, no direct biological rationale is present. Our concern is since they have developed stroke, they should have some underlying causes of developing stroke. Thus, we are faced with inferring a cause from the result. This process of inference is called Bayesian inference [4]. The fact that such a Bayesian factor is a major predictor raises a question. If the presence of prior stroke is so important, what is the influence of the absence of stroke? It is possible to speculate that the

2 446 T. Nakamizo, M. Yamamoto longer a patient is stroke-free, the lower his (or her) stroke risk actually is. This subject has remained unaddressed yet. It requires long-term observation in a large number of patients to answer the subject clinically. In this paper, we made an attempt to evaluate the above hypothesis. We modeled the development of stroke as a yearly Bernoulli trial, assuming that whether a patient develops stroke in a year is determined at the beginning of the year. Applying a conventional Bayesian inference method to the model with correction for the risk increase by aging, we simulated how stroke-free duration could affect the estimated stroke risk in patients with AF. Methods Data in the meta-analysis by Stroke Risk in Atrial Fibrillation Working Group [1] was used for this study. This meta-analysis presented the combined relative risk of the predictors, summarizing seven prospective studies for investigating independent predictors of stroke in non-anticoagulated patients with AF. Bayesian inference We included TIA into the term stroke, and estimated the stroke risk in a patient who has been stroke-free for n years. The development of stroke was modeled as a yearly Bernoulli trial. Namely, we assumed that whether a patient develops stroke in the year n+1 is determined at the beginning of the year, where the probability of developing stroke in that year is p n, and the probability of non-developing stroke is (1-p n ). This model includes factors of age and prior stroke alone, assuming that other risk factors are unchanged. We applied a conventional method of the Bayesian inference [4] to the model, assuming the Beta prior distribution. As shown in Appendix, the probability, p n, for stroke to occur in the year n+1 is expressed as follows: n /10 p n = p0r. (1) n /10 1/10 1+ p0( λ 1)( R 1) /( R 1) In the above, λ is the relative risk of prior stroke, R is the relative risk of aging expressed as times per decade, and p 0 is the prior probability that stroke occurs in the first year. The above probability, p n, corresponds to the annual risk of stroke if a patient does not develop stroke for n years, provided that other risk factors are unchanged. Simulation In equation (1), we have two parameters, λ and R, the relative risk of prior stroke and that of aging (expressed as times per decade). The meta-analysis [1] estimated λ=2.5 (95% confidence interval [CI]; ) and R=1.5 (95%CI; ), respectively [1]. In the present simulation, we treated and R as random variables, and assumed that these variables have a log-normal distribution characterized by the probability density function: 1 (log x μ) exp [- 2 2πσx 2σ where x stands for λ or R, μ and σ are the parameters to characterize the distributions. The parameters μ and were determined using a simulation software Crystal Ball, (Decisioneerin, Inc, Denver, USA), such that the 95%CI coincided with those reported in the metaanalysis [1] (Table 1). The resultant point estimates (represented by the mode) were 2.44 and 1.48 for λ and R, respectively. These values were minimally deviated from those estimated in the meta-analysis. We performed simulation for years from n = 1 to 10 and for various levels of the initial risk: p 0 = 1,, 8 %/year. For each n and p 0, after randomly generating values of λ and R 2,500,000 times according to the above distribution (2), we noted the mode and the 95% CI of the stroke risk given by equation (1). Results Figure 1 shows the result of the present simulation for patients with AF. If stroke-free duration were ignored, stroke risk should increase by 1.5 (95% CI, ) times per decade. Apparently, stroke-free duration reduced the estimated stroke risk, offsetting the risk increase by aging. In patients with low-to-moderate initial risk, the risk reduction by increasing stroke-free duration was approximately equal to the risk increase by aging. The estimated risk remained almost constant with aging when they were stroke-free. For example, suppose a patient with stroke risk 3%/year (estimated) does not develop stroke for ten years. Then, his (or her) stroke risk is calculated as 2.9%/year (95%CI, %). On the other hand, in patients with high initial estimated risk, the risk reduction by increasing strokefree duration was greater than the risk increase by aging. The estimated risk decreased when they were stroke-free. For example, if a patient whose estimated stroke risk is 8%/year does not develop stroke for 10 2 ], (2)

3 Vol. 3 No. 4 August 2009 Stroke-free duration in atrial fibrillation 447 Table 1. The parameters of the distribution for the relative risk of prior stroke (λ) and aging (times per decade) (R). Parameters Representative values 95%CI μ σ mean median mode lower limit upper limit λ R years, the stroke risk becomes 4.7%/year (95%CI, %). Discussion Our model predicted that the stroke risk in patients with AF might be reduced by increasing stroke-free duration up to equal or larger degree, compared with the risk increase by aging. This suggests that strokefree duration is an important predictor of freedom from stroke, as important as aging. The most critical assumption of our model was that prior stroke or TIA should be a Bayesian predictor. Since stroke risk cannot be measured in a patient, the risk has to be estimated with epidemiological data. Therefore, stroke risk in a patient is regarded as subjective probability [4], because it is the degree of our belief, made by estimation, on how likely the patient will develop stroke. This notion first allows us to use the Bayesian inference method when dealing with stroke risk in a patient. Next, it is not biologically plausible to assume that prior stroke causes future stroke, but prior stroke merely makes us infer that the patients have some pathological condition to promote the development of stroke. This process of inference, in turn, justifies our assumption that prior stroke or TIA should be a Bayesian predictor. The second assumption of our model was that the development of stroke should be described as a yearly Bernoulli trial. This simplification may be justified if the probabilistic property of developing stroke is sufficiently preserved. The third assumption of our model was that the stroke risk in a patient should be expressed by a Beta distribution. Since the stroke risk in a patient is regarded as a subjective probability, its distribution is not unique, being dependent on our belief. Therefore, this assumption is inevitably arbitrary, but it may be permissible when our interest is in estimating the mean value, not the distribution itself. It is usual practice to assume the Beta prior distributions in the Bayesian inference. In fact, the beta distribution is recognized as the natural conjugate distribution of a Bernoulli process [4]. It is controversial to set the threshold of stroke risk when anticoagulation should be started in patients with AF but without prior stroke. The ACC/AHA/ ESC guideline [5] does not recommend anticoagulation for patients without risk factors, but recommends it for those with two or more risk factors. For patients with one risk factor, the guideline states that anticoagulation may be reasonable. The CHADS 2 score [2] is often used for stroke risk stratification in patients with AF. This scheme assigns each risk factor a value of one or two. If this scheme is used, the threshold of stroke risk will be between CHADS 2 score zero and two. Patients with CHADS 2 score 0, 1, or 2 have annual stroke risk (including TIA risk) of 1.9%, 2.8%, or 4.0%, respectively, as shown in Table 2 and also shown as the dotted lines in Fig. 1. It is apparent from Fig.1 that the risk reduction by increasing stroke-free duration is larger in highrisk (4%/year or more, including TIA risk) patients, but the posterior risk is still high enough to warrant anticoagulation. Therefore, in patients with high risk, increasing stroke-free duration will not affect our decision to start anticoagulation. However, we must note that the posterior risk may go below the threshold under the following conditions: 1) we set a higher threshold, 2) the initial risk is lower, 3) the risk takes a lower path, and 4) patients are stroke-free in a sufficiently long period. In patients with low-to-moderate-risk (3%/year or less, including TIA risk), the stroke risk remained relatively constant as long as they are stroke-free. The risk did not overtly cross the dotted lines in Fig. 1. This suggests that the initial decision should hold against aging so long as the patient is strokefree. Therefore, so long as a patient with AF is strokefree, it may not be appropriate to start anticoagulation only for the reason that he (or she) becomes 75 years

4 448 T. Nakamizo, M. Yamamoto Fig. 1 The change of stroke risk after stroke-free duration in patients with atrial fibrillation (AF). The plots represent the simulated annual stroke risk including transient ischemic attack (TIA) risk in patients who do not develop stroke or TIA for 1,.,10 years. Each graph represents the annual risk starting from the initial risk p 0 =1,.,8%/year. The initial risk is the stroke and TIA risk estimated at the beginning of the first year. The bold curve is the point estimation represented by the mode, while the upper and lower solid curves are the upper and lower limits of the 95%CI, respectively. The dotted lines are, from the bottom to the top, 1.9%, 2.8%, and 4.0%, which correspond to the stroke and TIA risk of CHADS 2 score 0, 1, and 2, respectively.

5 Vol. 3 No. 4 August 2009 Stroke-free duration in atrial fibrillation 449 Table 2. CHADS 2 risk stratification scheme [2]. This scheme assigns a score of one or two to the risk factors for stroke in patients with atrial fibrillation. Total CHADS 2 score in a patient is used to estimate his (or her) stroke risk. Risk factors Score Recent congestive heart failure 1 Hypertension 1 Age (75 years or older) 1 Diabetes mellitus 1 History of stroke or transient ischemic attack 2 CHDS 2 score Stroke risk (95%CI) (%/year) ( ) ( ) ( ) ( ) ( ) ( ) ( ) old. For example, let us consider an example of a 70- year-old woman with lone AF. According to the CHADS 2 index [2], her estimated stroke risk (including TIA risk) is 1.9%/year. In this case, anticoagulation is not recommended [5]. Suppose that she has not developed stroke or TIA for the next five years. According to the ACC/AHA/ESC guideline [5], a woman with AF over 75 years of age should be anticoagulated. Then, should you now start anticoagulation on her? Our simulation result shows that in patients with low-to-moderate risk, the estimated stroke risk remains almost constant with aging if they remain stroke-free (Fig. 1). Therefore, anticoagulation may not be appropriate if she actually has not developed stroke for five years. Thus, physicians must take into account stroke-free duration when deciding whether to start anticoagulation in individual patients. The present study is based on a theoretical prediction. To validate the model, we need clinical data on stroke risk during long-term observation in many patients with AF. There have been no clinical studies to include such data, but our prediction may be useful as a predictor of future stroke. Conclusion Our analysis predicted that stroke-free duration in patients with AF is an important predictor of freedom from stroke. Increasing stroke-free duration will reduce the estimated stroke risk, affecting our decision to start anticoagulation. Therefore, strokefree duration has to be considered when estimating stroke risk in patients with AF. The authors have no conflict of interest to declare. Appendix. Development of the Bayesian inference Let us assume that the prior probability for stroke to occur in the first year is expressed by a Beta distribution, Beta (a 0, b 0 ). Then, the prior mean probability of developing stroke in the first year is given as p 0 = a 0 ). First, let us apply the Bayes theorem [4]. Then, we find that the posterior distribution is expressed as: Beta (a 0, b 0 +1), if stroke does not occur in the first year, Beta (a 0 +1, b 0 ), if stroke does occur in the first year. Consequently, the posterior mean probability is expressed as: a 0 +1), if stroke does not occur, (a 0 +1) +1), if stroke does occur. Next, let us incorporate aging into the model. According to the meta-analysis [1], aging is associated with an incremental increased stroke risk of 1.5 per decade (95%CI: ). If this value is denoted by R, stroke risk increases by r = R 1/10 times per year. Therefore, the probability for stroke to occur in the second year is expressed as: p 1 = ra 0 +1), if stroke does not occur in the first year, p 1, = r(a 0 +1) +1), if stroke does occur in the first year. According to the meta-analysis [1], the relative risk of prior stroke or TIA was 2.5 (95%CI: ). Let

6 450 T. Nakamizo, M. Yamamoto us denote this value by the symbol λ. In the studies included in this meta-analysis, the average number of prior strokes or TIAs per patient was unknown, but it is expected to be approximately one. Thus, we here assume that developing stroke once increases stroke risk λ times. Then, it follows that p 1 /p 1 =λ. In virtue of this relationship, we obtain the relationship between a 0 and λ as a 0 = 1/(λ-1). If stroke does not occur in the first year, the prior mean probability in the second year is expressed as p 1 = ra 0 +1). Here, we assume the prior distribution to be Beta (a 1, b 1 ). Then, the prior mean probability in the second year is p 1 = a 1 /(a 1 +b 1 ) = ra 0 / (a 0 +1). By applying the Bayesian inference, the probability of developing stroke in the third year is p 2 = ra 1 /(a 1 +b 1 +1), if stroke does not occur in the second year, p 2 = r(a 1 +1)/(a 1 +b 1 +1), if stroke does occur. From the above procedure, we find that a 1 =1/(λ-1). For any n, a n = 1/(-1), and let us denote this constant by the symbol a. Extension of this procedure leads to the recurrence equation as follows: p n = a/(a+b n ) = ra/(a+b +1), if stroke does not n-1 occur in the year 1,,n. Here, p n is the probability that stroke occurs in the year n+1, the parameters a = 1/(λ-1), and b n are those of the prior distribution Beta (a, b n ) in the year n+1. The constant r = R 1/10 is the annual rate of increase in the stroke risk. The recurrence equation was solved to give equation (1) in Methods, if stroke does not occur in year 1,,n. References 1. Stroke Risk in Atrial Fibrillation Working Group. Independent predictors of stroke in patients with atrial fibrillation: a systematic review. Neurology. 2007; 69: Gage BF, Waterman AD, Shannon W, Boechler M, Rich MW, Radford MJ. Validation of clinical classification schemes for predicting stroke: results from the national registry of atrial fibrillation. JAMA. 2001; 285: Stroke Risk in Atrial Fibrillation Working Group. Comparison of 12 risk stratification schemes to predict stroke in patients with nonvalvular atrial fibrillation. Stroke. 2008; 39: Winkler RL. An Introduction to Bayesian Inference and Decision. 2nd. ed. Gainesville (Florida): Probabilistic Publishing; Fuster V, Rydn LE, Cannom DS, Crijns HJ, Curtis AB, Ellenbogen KA, et al. ACC/AHA/ESC 2006 guidelines for the management of patients with atrial fibrillation. Circulation. 2006; 114: e