The Bleeding Risk and Natural History of Idiopathic Thrombocytopenic Purpura in Patients With Persistent Low Platelet Counts

Background  No firm data are available on the natural history of idiopathic thrombocytopenic purpura (ITP) or on mortality rates or frequency of major bleeding episodes associated with this condition. The disease is thought to have a relatively benign course, despite the frequent occurrence of very low platelet counts. This prevailing conception often guides therapeutic decisions.

Objective  To estimate the bleeding risk of ITP involving persistent low platelet counts (<30 × 10⁹/L) and its impact on prognosis.

Design  Age-adjusted bleeding risk was derived from a pooled analysis of ITP clinical series based on a systematic literature search. The risk estimate was incorporated into a Markov model to determine its impact on prognosis.

Results  Seventeen case series complied with inclusion criteria, including 1817 patients with ITP. There were 49 cases of fatal hemorrhage over an estimated 1258 to 3023 patient-years at risk. The rate of fatal hemorrhage before age adjustment was estimated at between 0.0162 and 0.0389 cases per patient-year. Age-adjusted rates were 0.004, 0.012, and 0.130 cases per patient-year for age groups younger than 40, 40 to 60, and older than 60 years, respectively. Predicted 5-year mortality rates ranged from 2.2% for patients younger than 40 years to 47.8% for those older than 60 years. A 30-year-old woman remaining thrombocytopenic due to ITP was predicted to lose 20.4 years (14.9 quality-adjusted life years) of her potential life expectancy. At age 70, predicted loss was 9.4 years (5.0 quality-adjusted life years).

Conclusions  Idiopathic thrombocytopenic purpura with persistent low platelet counts carries a grave prognosis. Therefore, an active therapeutic approach in the clinical management of affected patients should be considered. In view of the significant potential implications of the model results, we call for initiating a well-designed prospective inception cohort study of patients with ITP.

ADULT idiopathic thrombocytopenic purpura (ITP) is a relatively common and easily recognizable bleeding disorder. Nonetheless, the natural history of this condition is unknown.¹ The course and prognosis of ITP is mainly determined by the risk of spontaneous bleeding associated with low platelet counts.^1,2 The American Society of Hematology (ASH) ITP guideline panel recently published their conclusions based on a systematic review of clinical series.² They estimated that the risk of fatal hemorrhage was approximately 5% throughout the lifetime of a patient with ITP. However, this estimate is an average risk obtained in a heterogeneous group of patients. It included both patients who had favorable responses to therapy, and those who remained refractory.

Furthermore, the estimate refers to patients with platelet counts below 30 × 10⁹/L, a level commonly accepted as the threshold for the occurrence of major bleeding events among patients with ITP,^1-3 as well as patients who maintained counts above this dangerous level. In addition, it does not account for the large variations in bleeding risk at different ages.⁴ Consequently, this composite estimate of mortality risk is of little value to the physician faced with a clinical management dilemma regarding a specific patient with ITP and persistently low platelet counts. Such dilemmas arise in patients refractory to first-line therapy or those at high risk for adverse effects. Hence, the consequences of withholding treatment must be weighed against immediate and long-term risks associated with this treatment.³

Knowledge of the natural history of ITP in terms of bleeding risk and decline in life expectancy (LE) and quality of life (QOL) could greatly contribute to the decision process. In this article, we attempt to estimate the prognosis of patients with ITP involving persistent low platelet counts (<30 × 10⁹/L) by (1) performing quantitative assessment of bleeding risk in ITP based on pooled data obtained from the literature and (2) determining the effect of this bleeding risk and remission rate on LE and quality-adjusted LE (QALE).

We considered patients with ITP to be at risk for major hemorrhage when their platelet counts fell below 30 × 10⁹/L.^1-3 The analysis referred to 4 bleeding categories: (1) fatal bleeding events; (2) hemorrhagic strokes with residual disability; (3) major "transient" bleeding events requiring hospitalization, but without long-term sequelae (eg, upper gastrointestinal tract bleeding); and (4) minor oozing and bruising (eg, gingival bleeding event). We assumed no difference in bleeding rates between the sexes.

We used the results of the extensive systematic literature search performed by the ASH panel for establishing practice guidelines for ITP.² This search was extended to include later publications (years 1995-1998) using a MEDLINE query (keyword: idiopathic thrombocytopenic purpura/ITP, all fields, and all subheadings, limited to articles in the English language, excluding letters and case reports). Candidate articles were systematically reviewed to assess their appropriateness for inclusion in this analysis according to the following criteria: (1) studies that observed cohorts of patients with ITP over a period of at least 1 year (either prospective or retrospective); (2) either the number of deaths due to hemorrhagic events or the absence of such fatalities within the study period was clearly reported; (3) the study period had been sufficiently characterized (either exact patient-years of observation reported or both maximum and median follow-up periods); and (4) the number of patients at risk for thrombopenic hemorrhage during the study follow-up period was reported (ie, patients with periods of platelet counts <30 × 10⁹/L).

The bleeding rate was calculated as the ratio between the pooled number of bleeding events and the pooled patient time at risk for major bleeding in the studies included in the analysis. The rates of fatal bleeding events and major nonfatal bleeding events were calculated separately. The calculation of the latter was based on the subset of articles that followed up and reported major, nonfatal bleeding events.

Patient time at risk for major bleeding events included the period prior to response to therapy in all patients and the follow-up period among refractory patients or patients who had experienced relapse. Time until response was estimated according to the treatment modality used: 2, 3, and 4 weeks for splenectomy, corticosteroid treatment, and cytotoxic therapy, respectively.

Estimation of the follow-up period for patients who did not respond to therapy or those who experienced a relapse was either (1) the exact period of follow-up, when reported, or (2) when the exact period was not stated, we estimated a plausible range between a high boundary maximum follow-up period in the study and a low boundary (the median follow-up period). We selected this low boundary based on the assumption that patients who remained thrombopenic tended to continue with medical attention for longer rather than shorter periods. For patients who experienced relapse, time until relapse was subtracted from the follow-up period.

Age adjustment of bleeding risk was based on the findings of Cortelazzo et al,⁴ who reported the age-associated risk for major bleeding in chronic ITP. They reported an odds ratio (OR) of 2.8 for patients aged 40 to 60 years vs those younger than 40, and of 28.9 for those 60 years or older vs those younger than 40 years, for a major bleeding event due to ITP. Considering the low absolute annual fatal bleeding risk, these ORs were used as estimates of the relative risks. We performed a subanalysis of the articles that reported the age distribution of their populations. Assuming linear relations, a set of equations was derived based on the age distribution of the pooled cohort, the relative risks in different age groups, estimated patient time in each age group, and the total number of bleeding events. From these equations, the age-adjusted risks within each of the above age groups were determined (Appendix).

The incidence of disabling stroke was not specified in any of the studies included in our analysis. Since this complication has major long-term effects on QOL, we used an indirect approach to estimate its frequency, based on the outcomes of hemorrhagic stroke from all causes (ie, not necessarily related to ITP): (1) The proportion of strokes among patients with ITP and fatal bleeding events was calculated based on the studies in our analysis that reported the site of bleeding (ie, central nervous system, gastrointestinal tract, etc). (2) According to a recent quantitative epidemiological review of intracerebral hemorrhage (summarizing 4 outcome studies),⁵ the fatality rate was 30% to 50%, and the rate of long-term disability among survivors ranged from 13% to 46%. Accordingly, we assumed 40% mortality and 30% (among survivors) long-term disability among patients with ITP who experienced a hemorrhagic stroke. (3) Estimation of the incidence of nonfatal hemorrhagic strokes among patients with ITP was based on the assumption that the fatality rate of ITP-related hemorrhagic strokes is similar to that of hemorrhagic strokes not associated with ITP. (4) The ratio of nonfatal to fatal strokes was estimated as 0.6:0.4; therefore, the rate of nonfatal hemorrhagic stroke in ITP equals 0.6 × R_FS/0.4, where R_FS indicates the rate of fatal stroke in patients with ITP with persistent low platelet counts.⁵ The rate of disability among hemorrhagic stroke survivors was estimated as 30%; therefore, the rate of nonfatal hemorrhagic stroke in ITP equals (0.3 × 0.6 × R_FS)/0.4, which equals 0.45 × R_FS.

A computerized Markov model was developed to simulate the course of a hypothetical cohort of patients with thrombocytopenia (platelet counts <30 × 10⁹/L) due to untreated or refractory ITP (Figure 1).^6,7 Initially, all cohort members are placed in the well-ITP state. At each cycle of the simulation (lasting 1 year), any cohort member may suffer a major hemorrhage that may be fatal, disabling, or transient. Accordingly, at the end of the cycle, members are transferred to dead or disabled states or remain in the well state. Major transient hemorrhagic events are assumed to exert a negative effect on ITP health states for 1 week. The probabilities of these events are determined according to the patient's age, based on the results of our pooled analysis. Patients also stand a chance of achieving a spontaneous remission, transferring into a remission state, or they may die from causes unrelated to ITP. The simulation continues until the entire hypothetical cohort has died. The QALE is calculated by summing up the number of patients in each state multiplied by the utility of that state, and dividing the sum by the cohort size. The average LE for cohort members is calculated similarly using a utility value of 0 for death and 1 year for all other health states. The predicted LE and QALE of patients with ITP are compared with those of a healthy cohort of the same age (a model including only well and dead states).

The model was constructed using DATA 3.0 software⁸ (TreeAge Software Inc, Williamstown, Mass). Analyses of women aged 30, 50, and 70 years with thrombocytopenia due to ITP (platelet counts <30 × 10⁹/L) were presented as base cases.

Model probabilities were based on the estimates from our pooled analysis^4,9-24 (Figure 2). Probabilities derived for age groups younger than 40, 40 to 60, and older than 60 years were affiliated with ages 30, 50, and 70 years, respectively. Linear interpolation was used to estimate bleeding probabilities for ages between these points.

Evidence is lacking for the rate of spontaneous remission, and although experts agree that this is infrequent, it was approximated to be 5% by the ASH panel.² In the model, we assumed a probability of 0.1% per year throughout the patient's lifetime (5% over 50 years).

The model considered death from any reason other than ITP, according to age and sex. Probabilities were based on the report of the National Center for Health Statistics.²⁵

Utility coefficients introduced QOL weightings for the various health states into the model. Patients in remission were considered equivalent to the general population, based on a survey using the Quality of Well-Being Index (QWB). The QWB was previously used to estimate utility values for the general population, adjusted to age and sex. The QWB scale derives its values from an assessment of patient answers to questions directed toward current symptoms and health-related reductions in mobility and physical and social activity. Scale scores range from 0 (death) to 1.0 (asymptomatic functioning).²⁶ Utility values for patients disabled by a hemorrhagic stroke were half of the utility value of healthy persons at the same age.²⁷ Patients with thrombocytopenia but without major bleeding have some reduction in their well-being score because of restrictions of engaging in sport activities, tendency toward bruising and minor bleeding events, anxiety of potential hemorrhage, and the need for repeated blood withdrawal. For these patients, we used a utility value of 0.98 of their healthy counterparts, based on the analogous state of bleeding tendency from anticoagulant therapy.²⁸ A utility value of 0 was used for a period of 1 week following events of major transient hemorrhage without long-term sequelae.

Sensitivity analysis was performed on all model estimates within plausible ranges within the DATA model. In addition, we used structural sensitivity analysis of assumptions regarding age adjustments. The model was modified to examine the effect of no age adjustment and different levels of relative risks among age groups. In addition, we examined the effect of using an analytic exponential equation based on the risk at 3 ages (30, 50, and 75 years) for calculation of age-adjusted risk.

To validate the model's predictions of survival unrelated to ITP, the probability of remission was set to 1.0 (bleeding rate drops to 0). The resulting LE was compared with those published by the National Center for Health Statistics.²⁵

Of the 295 articles found in the ASH search and the 293 found in our MEDLINE extensions, 17^4,9-24 (Table 1) complied with our inclusion criteria. These studies included a total of 1817 patients with ITP. There were 49 cases of fatal hemorrhage. The total patient time at risk for fatal bleeding events was estimated between 1258 and 3023 patient-years (low and high patient time estimates); accordingly, the annual fatal bleeding rate (without age adjustment) was estimated to be between 0.0162 and 0.0389 cases per patient-year (Figure 3).

Twelve of the 17 studies reported the anatomic site of bleeding.^{4,13,15-17,19,21,24-28} Among these studies, 31 of the fatal bleeding events were intracranial and 8 were at other sites; thus, the proportion of central nervous system bleeding events among fatal bleeding events was 31 of 39 (0.79). Nine of the 17 included studies that reported the age distribution of their cohorts^{4,9,10,12-14,17,19,24}: these studies included 34 of the 49 fatalities. Subanalysis of these 9 studies yielded risk for fatal bleeding event ranges between 0.4% per year for patients younger than 40 years and up to 13% per year for patients older than 60 years (Figure 3, left).

Estimation of the risk for major, nonfatal hemorrhage was based on 2 studies (with relatively elderly cohorts)^4,12; 29 events were reported within an observation time ranging from 77 to 105 patient-years. Age-adjusted risk for an event of nonfatal major hemorrhage was found to be 3% (0.03) per year for patients younger than 40 years, and 71% (0.71) per year for patients older than 60 years (Figure 3, right).

Figure 4, top, shows the cumulative probability for fatal bleeding for periods between 6 months and 5 years for each of the age groups analyzed. Predicted 5-year fatality ranges from 2.2% for patients younger than 40 years to 47.8% for patients older than 60 years. Seventy-six percent of the patients older than 60 years who remain with persistent low platelet counts will undergo at least 1 major nonfatal bleeding event during 2 years of follow-up (Figure 4, bottom). A 30-year-old woman with low platelet counts stands a 16.3% chance of a fatal bleeding event if left untreated over a period of 20 years, and a 73% chance of experiencing at least 1 major bleeding event.

Figure 5 presents the effect of untreated ITP on LE according to the age of onset. A 70-year-old woman is predicted to lose 5 quality-adjusted life years (QALY) (9.4 years), whereas a 25-year-old woman remaining thrombopenic owing to ITP will lose approximately 15 QALY (20.4 years) of her potential LE (the difference between life years and QALE is mainly attributable to reduced QOL weights in later years of life rather than the effect of ITP).

Table 2 gives results of the sensitivity analyses on all model estimates. Implementation of low and high bleeding risk estimates yielded predicted QALEs of 25.06 and 19.89 years, respectively, compared with the base case analysis of 21.97 years for a 30-year-old woman with ITP and 36.21 QALY for a healthy 30-year-old woman. The model was tested using a constant bleeding rate of 0.0288 events per patient-year (point estimate without age adjustment; Table 1) and also with less extreme differences in the estimates for relative risk of bleeding among age groups. This resulted in worse predicted prognosis compared with the base case (17.69-21.11 QALY). Variations of other model parameters caused only minor changes in model-predicted QALE.

Validation of the model using a remission probability of 1.0 (ie, no bleeding events) yielded LEs identical to our modeling results of the healthy cohort. These predictions were within 3 months of those reported by the National Center for Health Statistics for the general population.²⁵

This study derived a quantitative estimate of the risks for major and fatal hemorrhage in patients with persistent low platelet counts due to ITP by pooling data from clinical case series. We found that the risk for fatal hemorrhage ranges from 0.4% per year for patients younger than 40 years to 13% per year for patients 60 years and older. By using a Markov state transition simulation model, we were able to estimate the predicted loss of LE and QALE due to untreated ITP characterized by low platelet counts. We found this loss ranged from 20 years (15 QALY) for a 25-year-old patient to 9 years (5 QALY) for a 70-year-old. Thus, our analysis implies that patients with ITP who remain with low platelet counts have a poor prognosis.

Our study is an effort to establish evidence-based conclusions regarding the natural history of untreated ITP with persistent low platelet counts. The validity of our conclusions depends on both the quality of the data used and the strengths and limitations of the analytical and modeling methodologies.

Selection of studies for inclusion in the pooled analysis was based on a systematic and extensive search using explicitly defined inclusion criteria. Since no well-designed studies on the natural history of ITP with low platelet counts were found in the medical literature, we resorted to pooling data from uncontrolled, mostly retrospective clinical series—a relatively weak form of clinical evidence (only 2 prospective series were found^22,23). Nevertheless, several indications support acceptability of the evidence from these series. All series used strict criteria for diagnosis of ITP; 16 of the 17 series stated the criteria explicitly (ie, low platelet count, megacariocytosis on bone marrow, absence of splenomegaly, and ruling out other causes, including drugs, autoimmune diseases, infection, etc).

The use of retrospective series gives rise to concern of selection bias, ie, that the series include a high proportion of patients with ITP who are at increased risk. Some reassurance against this possibility is provided by the relative uniformity observed between the study estimates. Except for 1 outlier (a study of ITP in the elderly), the estimates of 16 individual trials from different settings were closely clustered around the pooled estimate (Figure 2). In addition, a subanalysis of 8 of the series, which included consecutive patients with ITP admitted to a health care center (either all patients with ITP or all those referred for splenectomy; Figure 2), revealed no difference in the resulting bleeding risk compared with the nonconsecutive series.

Another potential concern is that some of the series are relatively old (from the 1960s and 1970s), suggesting poorer prognosis compared with that of patients undergoing current treatment owing to less advanced medical care. Yet, as shown in Figure 2 in which the series are ordered chronologically, no such trend was observed. While these observations do not guarantee that the pooled estimates from these series would be equivalent to prospectively designed studies, we believe that they provide valuable reassurance supporting the accuracy of these estimates.

We used data pooling and Markov modeling to estimate bleeding risk and its implications on patient prognosis, respectively. By applying these analytical methods we were able to model the natural course of ITP in patients with persistently low platelet counts.

These methods enabled us to discriminatively simulate the health life course of a specific group of patients with ITP, ie, those patients among the series who remain with low platelet counts. We were also able to adjust for the increased risk at advanced age. This focus on high-risk patients is important because the ITP population is quite heterogeneous. It includes patients whose thrombocytopenia is not within severe levels (ie, >30 × 10⁹/L), those who responded favorably to treatment, and those whose follow-up period did not extend through advanced age. Thus, the prognosis of an "average" patient with ITP would seem to be only mildly affected. This might explain the discrepancy between the prevailing conception that ITP has a relatively benign course³ and the poor prognosis implied by our analysis.

Modeling the natural course of ITP required the use of several estimates. The stability of the models' results considering these uncertainties was verified by employing extensive sensitivity analysis, ie, examining the effect of varying these estimates along their plausible ranges.

Since the exact period of patient observation was not reported in most studies, we used an average between 2 extreme patient time estimates. When the model is set to the low estimates for bleeding risk, a loss of 11.15 QALY is predicted (36.21−25.06); setting high estimates yields a loss of 16.32 QALY (36.21−19.89) (Table 2). According to the base case analysis, a 30-year-old patient with ITP is predicted to lose 14.24 QALY (36.21−21.97).

The quantitative estimation of the excessive risk for major bleeding among the elderly, relative to young patients, was based on the study by Cortelazzo et al.⁴ Another series that included a large elderly cohort¹² supported this finding: a rate of 0.2 fatal bleeds per patient-year was found, compared with 0.016 to 0.042 per patient-year in the general pooled analysis (Table 1). When a constant bleeding rate was assumed (0.028 fatal bleeding events per patient-year), the model predicted a loss of 18.02 QALY. When less extreme differences in relative bleeding risks between age groups were used, the predicted prognosis was found to be worse than the basic model assumption, consisting of losses of 15.1 to 18.5 QALY. The reason is that by shifting the risk to a younger age, more events will happen earlier in life when they have a greater impact on remaining LE. Thus, assuming a relatively large effect of advanced age on the bleeding risks was a conservative assumption.

Finally, our age adjustment calculations were based on the relative risks in 3 age groups only, an insufficient number of points for reliable analytic curve fitting. We therefore approximated by using linear interpolations, though the true effect of age is probably not linear. Using an exponential curve instead of linear interpolation had only a small effect on the results (Table 2). Thus, varying the bleeding risks between the extreme conceivable estimates based on our data pooling, as well as modifying our assumptions on age adjustment, does not change our main finding, ie, a substantially worsened prognosis due to ITP with low platelet counts.

Additional estimates of probabilities and utilities in the model were based on extrapolations from analogous scenarios in patients without ITP (eg, outcomes of hemorrhagic stroke in the general population). It should be emphasized that our findings concerning the grave prognosis of ITP in terms of compromised LE are completely independent of these extrapolations. The extrapolated estimates are used solely for allowing supplementary calculations of outcomes in terms of QALY. These estimates were obtained from the published medical literature. This extrapolation might have introduced some inaccuracy; however, there is no better evidence available. To examine the effect of potential inaccuracies in these parameters, we examined them over a wide range in sensitivity analyses. The effect of varying these parameters within a plausible range was minimal (Table 2).

The results of our analysis are quite surprising, considering the prevailing conception that ITP has a relatively benign course.³ While the LE of an "average" patient with ITP seems to be only mildly compromised, our model indicates that this might not apply for patients with persistently low platelet counts. Considering the consistency of the evidence and the stability of the models' results despite uncertainties in the estimates used, we believe that these conclusions regarding the poor prognosis of ITP with persistently low platelet counts should not be ignored. Because there are multiple effective interventions available for the treatment of ITP, this new evidence of poor prognosis should strongly influence the aggressiveness of clinical management. This issue is of special importance in cases where the benefits of intervention are weighed against substantial adverse effects or risks, ie, high operative risk for splenectomy or corticosteroid dependency with associated adverse effects.

In this article, our main purpose was to estimate the quantity of LE loss due to ITP, and we did not particularly focus on modeling the effect of any specific therapy or sequence of therapies on ITP. Nevertheless, our model provides some insight about the potential effects of these treatments on LE and QALE in ITP. These effects can be achieved by modifying the probability for remission in the current model. We examined the effect on LE and QALE when this probability is set to 64%, which was estimated as a lifetime chance of recovery in ITP (spontaneous or in response to therapy) by George and Raskob.¹ We assumed this remission occurs throughout the first 2 years of model simulation. Under these conditions, the model predicted that a 25-year-old female patient with ITP will lose only 5.4 QALY (40.3−34.9) and 7.4 years of life (54.5−46.9), compared with a predicted loss of 14.9 QALY and 20.5 years without intervening in the natural history of the disease. This is a slight overestimate of the treatment benefits, because the effect of the treatment complications on LE and QALE was not modeled. However, it seems that based on the bleeding risks derived in this study, the effect of treatment on patients with ITP has a major favorable impact on the course of the disease.

According to our model, patients with ITP and persistent low platelet counts have a poor prognosis. There is a discrepancy between these conclusions and the prevailing conception regarding the benign course of ITP. Thus, the model would suggest taking an active therapeutic approach in the management of ITP, perhaps more active than that currently practiced. Inherent limitations in the quality of existing evidence (retrospective series) prevent definitive management recommendations; however, we believe that multiple retrospective ITP series, which converge toward a consistent estimate, provide evidence that should be considered when making clinical decisions. In view of the significant potential implications of the model results, we call for initiating a well-designed prospective inception cohort study of patients with ITP, so that the precise incidence of major bleeding can be discovered.

The age-adjusted risk for fatal bleeding among patients with a platelet count lower than 30 × 10⁹/L due to ITP was estimated according to the following considerations:

• Relative risks for major bleeding event in patients with chronic ITP, at age groups younger than 40, 40 to 60, and older than 60 years were based on the findings of Cortelazzo et al⁴: (1) BR_{age 40-60}/BR_{age <40}=2.8 and (2) BR_{age >60}/BR_{age <40}=28.9, where BR _{age group} indicates the bleeding risk of patients with ITP within that age group.

• Duration of follow-up period was assumed to be independent of age.

• The likelihood of a patient with ITP to be thrombopenic at a given moment was assumed to be independent of this patient's age. That is, a constant proportion of the patient observation time, r, comprises the "at-risk patient time" (time when platelet count <30 × 10⁹/L) in the 3 age groups: (3) PY_{age group}=r × N_{age group}, where PY _age-group indicates the patient time at risk of that age group; N _age-group, the number of patients in the age-group; and r, a constant. (The last 2 assumptions introduce some bias; however, they were unavoidable in performing age adjustment based on the available data. We accounted for this bias in the model by allowing wide ranges for sensitivity analyses.)

• Studies that reported the age distribution of the cohort were selected for this subanalysis. The total number of fatal bleeding events in these studies was determined. We assumed that the patient time contributed by each age group was proportionate to the relative size of each such group: (4)

For 3 possible age groups (<40, 40-60, >60 years). PY_{age group} indicates the patient time at risk contributed by each age group, the unknowns of our equations; PY_Total, the total patient time (within the pooled studies), and N, the number of patients.

• Finally, the bleeding risk within each age group was calculated as the ratio between fatal bleeding events and patient time of each group: (5) BR_{age group}=BE_{age group}/PY_{age group} and (6) BE_total=ΣBE_{age group}, where BR_{age group} indicates the bleeding risk adjusted for the age group; BE_{age group}, the number of fatal bleeding events of this age group; and PY_{age group}, the patient time at risk contributed by the age group.

Solving the set of linear equations 1 through 6 yields the age-adjusted absolute risks BR_{age group} for age groups younger than 40, 40 to 60, and older than 60 years.

Accepted for publication June 29, 1999.

We are grateful to Lawrence Freedman, PhD, for statistical consultations and thoughtful comments on the manuscript.

Reprints: Yael Cohen, MD, Gertner Institute for Epidemiology and Health Policy Research, The Chaim Sheba Medical Center, Tel Hashomer, Israel (e-mail: coheny@post.tau.ac.il).