Piecewise Analysis of Patient Survival after Onset of AKI

Summary

Background and objectives

AKI affects approximately 2%–7% of hospitalized patients and >35% of critically ill patients. Survival after AKI may be described as having an acute phase (including an initial hyperacute component) followed by a convalescent phase, which may itself have early and late components.

Design, setting, participants, & measurements

Data from the Veterans Affairs/National Institutes of Health Acute Renal Failure Trial Network (ATN) study was used to model mortality risk among patients with dialysis-requiring AKI. This study assumed that the mortality hazard can be described by a piecewise log-linear function with change points. Using an average likelihood method, the authors tested for the number of change points in a piecewise log-linear hazard model. The maximum likelihood approach to locate the change point(s) was then adopted, and associated parameters and standard errors were estimated.

Results

There were 1124 ATN participants with follow-up to 1 year. The mortality hazard of AKI decreased over time with inflections in the rate of decrease at days 4, 42, and 148, with the sharpest change at day 42. The daily rate of decline in the log of the hazard for death was 0.220 over the first 4 days, 0.046 between day 4 and day 42, 0.017 between day 42 and day 148, and 0.003 between day 148 and day 365.

Conclusions

There appear to be two major phases of mortality risk after AKI: an early phase extending over the first 6 weeks and a late phase from 6 weeks to 1 year. Within the first 42 days, this can be further divided into hyperacute (days 1–4) and acute (days 4–42) phases. After 42 days, there appear to be early (days 42–148) and late (after day 148) convalescent phases. These findings may help to inform the design of AKI clinical trials and assist critical care physicians in prognostic stratification.

Introduction

AKI is a heterogeneous condition that encompasses acute intrinsic renal failure, prerenal azotemia, and obstructive uropathy. AKI is a common complication of acute illness, affecting approximately 2%–7% of hospitalized patients (1–4) and >35% of critically ill patients (5–7). Many clinical trials in patients with AKI have been conducted over the past 20 years, largely aiming to improve the distressingly poor outcomes associated with the condition. In most cases, the primary outcome of interest was all-cause mortality. The timing of the assessment of mortality has varied, with reports of in-hospital mortality or at specific time points ranging from 28 days to 90 days. This variability in the duration of follow-up reflects the paucity of data regarding the duration of mortality risk associated with AKI. To date, no rigorous statistical analysis has been conducted to examine the most appropriate duration of follow-up despite the relatively large number of completed AKI trials.

Based on the natural history of AKI, we hypothesized that mortality after AKI might exhibit acute and convalescent phases, with the earlier consisting of hyperacute and acute components and the latter of convalescent (subacute) and late convalescent phases. In biomedical research, issues of change points arise when the survival and/or the hazard function of an event of interest experiences abrupt change(s) at defined time point(s). In AKI studies, rapid changes in hazard could occur between the acute phase and the convalescent phase, and/or between hyperacute and acute or an early and late convalescent phase. To test our hypothesis, we analyzed survival data from the Veterans Affairs (VA)/National Institutes of Health (NIH) Acute Renal Failure Trial Network (ATN) study to assess for the presence of change points in the survival curve.

Materials and Methods

Background of the ATN Trial

The VA/NIH ATN study (ClinicalTrials.gov, NCT00076219) was a multicenter randomized trial comparing strategies of intensive and less-intensive renal replacement therapy (RRT) in critically ill patients with AKI, that was conducted between November 2003 and July 2007 at 27 VA and university-affiliated medical centers in the United States (8,9). The study enrolled a total of 1124 patients. Patients were randomized within 24 hours after the initiation of RRT and received protocol-directed RRT for up to 28 days. The primary study endpoint was 60-day all-cause mortality. After 60 days, surviving patients were followed for up to 1 year for all-cause mortality. There were a total of 720 deaths (64%) and 28 patients (2%) were lost to follow-up by the end of 1 year. The two treatment arms had similar baseline characteristics. The study was approved by the Human Rights Committee at the West Haven Veterans Affairs Cooperative Studies Program Coordinating Center and by the institutional review boards at each of the participating study sites. Written informed consent was obtained from all patients or their health care surrogates, and patients who required surrogates were re-consented if they regained decision-making capacity. Further details of the study design and results were previously published (8,9).

Data

We combined data from the two treatment arms for the current analyses, because there was no treatment difference, which is evident from the Kaplan–Meier plot of the two treatment groups in Figure 1.

Figure 1.

Survival of all-cause mortality by treatment.

Adjustments were made to more precisely define the time from randomization to death. Assessment of survival time based only on date of randomization and date of death introduced the possibility of large biases in individuals with short survival times. Before adjustment, the surveillance time on day 1 was <12 hours for patients who were randomized in the afternoon, and the majority of patients in this study were recruited in the late afternoon due, at least in part, to the practicalities of obtaining surrogate consent. Therefore, we reduced the survival time of patients randomized between noon and midnight by 1 day from the original data. Using this adjustment, the actual range of survival for day 1 was restricted to a minimum of 0 and a maximum of 36 hours. As a result of this adjustment, 55 patients were reclassified from a survival time of 2 days to a survival time of only 1 day, and the count of patients who died within 1 day of randomization increased from 17 to 72 patients, highlighting the potential errors introduced by counting by calendar day without consideration of the time of day when randomization occurred.

Outcome

The outcome of interest in the current analysis was all-cause mortality. All participants were followed up to 365 days.

Statistical Analyses

Through visual evaluation, the observed hazard function of mortality in the ATN study appeared to be piecewise log-linear (10). The discrete hazard function h(t) denotes the probability of a patient dying on day t, given that the patient has survived to the beginning of day t. The log-linear hazard model sets h(t) = exp(a+bt). The slope b is permitted to take different values b₁, b₁₊b₂,…, b₁₊b_{2+ …+}b_k between the change points T₁, T₂,..., T_k, where k is the total number of change points. This model is a general linear model in the binomial family with log as link function. The changing slopes are estimated as increments of the original slope.

We applied the average likelihood ratio test (11) to test sequentially for the validity of new change points. For one change point as compared with no change point, the average likelihood ratio is computed by averaging the likelihood ratio over all possible change point days, weighted by the numbers of deaths occurring on each day. The significant values of the average likelihood ratio test are close to the standard asymptotic likelihood theory values that we can use for hypothesis test, unlike the likelihood ratio test itself (11,12). The detail of tests and significant values are provided in the Supplemental Material.

We estimated the change point by maximum likelihood when its existence was established in the previous testing step. For the first change point, all observed death days are considered as possible change points. The point that gives the maximum likelihood was the estimated change point.

Once the first change point was established, we tested for the validity of the second change point given the first change point. If the test was not significant, we would conclude there is only one change point. Otherwise, we would estimate the second change point by maximum likelihood. Both the testing and estimation procedures are similar to those for the first change point. The procedure was then repeated until the test of a new change point failed to reject the null hypothesis. For the ATN data, the test for the fourth change point turned out to be insignificant. We therefore propose a three-change-point log-linear hazard model. The 95% confidence intervals of change points were constructed by bootstrapping.

The program we developed and used for testing and estimation in R (13) is available upon request.

Results

Overall, there were 1124 participants with mean age of 60 years (SD ±15), and included 71% men and 74% Caucasians. Table 1 outlines the basic demographic and clinical characteristics at time of enrollment.

Table 1.

Demographics and clinical characteristics of the study participants

Characteristic	All Participants (n=1124)
Age (yr)	59.7±15.3
Male sex	793 (71)
Race/ethnicity
White	835 (74)
Black	179 (16)
Hispanic	77 (7)
Other	16 (3)
Premorbid renal function
Serum creatinine (mg/dl)	1.1±0.4
eGFR (ml/min per 1.73 m2)
≥60	686 (61)
45–59	237 (21)
30–44	124 (11)
Etiology of AKI
Ischemic	871 (77)
Nephrotoxic	286 (25)
Multifactorial	626 (56)
Postsurgery	463 (41)
Sepsis	579 (52)
Mechanical ventilation	905 (81)
Oliguria	877 (78)
SOFA score
Respiratory	2.3±1.1
Coagulation	1.4±1.2
Liver	1.4±1.3
Cardiovascular	2.3±1.7
Central nervous system	2.5±1.4
Totala	13.8±3.9
Cleveland Clinic score	12.1±3.4
APACHE II	26.4±7.3

Values are means ± SD or n (%). n is the total number of patients randomized. Percentages are based on patients without missing data. SOFA, sequential organ failure assessment; APACHE II, Acute Physiology and Chronic Health Evaluation.

^aAll patients assigned a score of 4 for renal component of the SOFA score.

In Table 2, we summarize the results of the sequential testing for change points. We initially compared a log-linear hazard model with one change point to a log-linear hazard model with no change point. The log average likelihood ratio was 113.4 with an associated P value of <0.001 and the estimated change point was at day 42. Thus, the presence of a first change point at day 42 is strongly supported. We then compared a log-linear hazard model with two change points to the model with only one change point. The log average likelihood ratio was 8.0 with an associated P value of <0.001. When we compared a log-linear hazard model with three change points to the model with two change points, the log average likelihood ratio was 4.8 and its associated P value was <0.001. These results support the presence of change points at day 4 and day 148, in addition to the first change point at day 42. When we compared a log-linear hazard model with four change points to the preceding model, the log average likelihood ratio was 0.16, its associated P value was 0.73. Thus we conclude that no more than three change points are needed with change points at days 42, 4 and 148. These change points also provide the maximum likelihood over all possible sets of three change points.

Table 2.

Average likelihood test for the validity of a new change point

Description	Hypothesis Tests
	1 Change Point versus 0 Change Points	2 Change Points versus 1 Change Point	3 Change Points versus 2 Change Points	4 Change Points versus 3 Change Points
Log average likelihood ratio	113.42	7.97	4.78	0.16
P value	<0.001	<0.001	0.001	0.73
Change point(s)	42	42, 4	42, 4, 148	42, 4, 148, 55

In Table 3, we summarize the three-change-point model: three change points, including their 95% confidence intervals, slope, and increase of slope in each phase and the hazard calculation. The first phase (day 0–4), representing the hyperacute phase of AKI, has a rate of decline in the hazard on log scale of 0.220 per day. The daily hazard for death is 0.05 on day one, decreases sharply to 0.03 by day 4. The second phase (day 4–42), represents the subsequent acute phase of AKI. The rate of decline in the hazard on log scale is 0.046 per day in this phase, which represents an increase of 0.174 from the slope of the first piece. The cumulative mortality during the hyperacute and subsequent acute phases (although day 42) is 46.4%.

Table 3.

Summary of the three-change-point model

Phase	Beginning Day (95% CI)a	Ending Day (95% CI)a	Description of Slopeb,c		Daily Hazard Function h(t)b,c
			log(h(0)) ± SE	−2.755±0.09
First (hyperacute)	0	4 (1 to 12)	Slope of log(h(t)) ± SE	−0.220±0.032	exp (−2.755 – 0.220*t)
Second (acute)	4 (1 to 12)	42 (22 to 66)	Slope increase of log(h(t)) ± SE	0.174±0.034	exp (−3.635 – 0.046*(t-4))
Third (early convalescent)	42 (22 to 66)	148 (100 to 180)	Slope increase of log(h(t)) ± SE	0.029±0.005	exp (−5.383 – 0.017*(t-42))
Fourth (late convalescent)	148 (100 to 180)	365	Slope increase of log(h(t)) ± SE	0.014±0.004	exp (−7.191 – 0.003*(t-148))

95% CI, 95% confidence interval.

^aThe beginning and ending days of each phase represent the change points at 4, 42, and 148 days.

^bThe value of t for each phase is restricted to the integer values of the days within each phase. In the first (hyperacute) phase, 0 ≤ t ≤ 4; in the second (acute) phase, 4 ≤ t ≤ 42; in the third (early convalescent) phase, 42 ≤ t ≤ 148; in the fourth (late convalescent) phase, 148 ≤ t ≤ 365.

^cThe slope of the first phase is −0.220; the slope of the second phase is −0.046, −0.220 + 0.174; the slope of the third phase is −0.017, −0.220 + 0.174 + 0.029; the slope of the fourth phase is −0.003, −0.220 + 0.174 + 0.029 + 0.014.

In Table 3, the third and fourth (“convalescent” and “late convalescent”) phases of AKI begin at roughly day 42. During the convalescent phase and late convalescent phase, the hazard for death continues to decline but at rates of 0.017 per day and 0.003 per day, respectively, on the log scale that are <1/13 of the rate of decline in the hyperacute phase. The daily hazard for death starts at 0.0046 at day 42 and ends at 0.00039 at day 365. The cumulative mortality in the convalescent and late convalescent phases is 15.6%.

Figure 2 displays the daily hazard plots of observed data and data fitted using the models with no, one, two, and three change points. The log likelihood values of no change point, one change point, two change points, and three change points are −279.4, −162.3, −152.3, and −143.8, respectively. Figure 3 shows the observed survival and the survival plots estimated by the models with no, one, two, and three change points. In both the survival and hazard plots, all three models with change point(s) show excellent fit to the observed data. The model with no change point does not appear to fit the observed data, whereas the model with two change points appears very similar to the model with three change points.

Figure 2.

Hazard of death under different models. The log likelihood values of the model with no change points or one change point (A) or two change points or three change points (B) are −279.4, −162.3, −152.3, and −143.8, respectively.

Figure 3.

Estimated survival under different models.

Discussion

By analyzing the change in the daily hazard for death using data from the ATN study, we are able to identify distinct phases in mortality risk after an episode of severe AKI. We identified three discrete change points in the ATN survival data, ranked in order of their significance or their sharpness in change, at day 42, day 4, and day 148. The log likelihood value improved from −279.4 to −162.3 by introducing the first inflection at day 42. The additional improvement in the log likelihood with introduction of change points at day 4 and at day 148 is <10, although they both demonstrate statistical significance. Similarly, there is dramatic improvement in the fitting of the plot of the daily hazard of death with the introduction of the first inflection point at day 42 (Figure 2). The improvement of fitting by including the second and third change points is not visually significant. Thus, one could describe the inflection at day 42 as a major change point, whereas the inflections at days 4 and 148 are minor change points. However, from the survival plots in Figure 3, it is apparent that the models with change points slightly overestimate the observed survival. This may result from the large number of days with no events, particularly after day 90, given the size of the study cohort. As a result, the overall intercept in the hazard may be underestimated. In addition, because the survival plot has an accumulating effect, any minor under fit at an early time would have greater effect at later time points.

These results can inform the selection of time points for assessing mortality in the design of clinical trials of AKI. Various time periods for assessment of mortality, ranging from 28 days to 6 months and including 35 days, 60 days, and 90 days, have been used in prior clinical trials (14–16). For example, in trials by Ronco and colleagues and Schiffl and colleagues (14,15), the primary study endpoint was all-cause mortality within 14 days after discontinuation of RRT. The ATN trial (8) used 60-day mortality as the primary mortality endpoint, which was believed long enough to cover the acute phase of AKI, whereas the Randomized Evaluation of Normal versus Augmented Level (RENAL) Replacement Therapy Study (17,18), completed in Australia and New Zealand, chose 90-day mortality as its primary outcome.

The ATN trial data provide accurate follow-up to 1 year after the onset of dialysis-requiring AKI. Using data from this well designed and implemented clinical study and utilizing advanced statistical methods, we provide a statistical analysis of the presence and length of the acute and convalescent phases of the course of AKI (19). We demonstrate that a model with one major change point and two minor change points provides a significantly better fit to the data then the models without a change point, or with only one or two change points. The primary inflection point, at day 42, divides the clinical course after an episode of AKI into acute and convalescent phases. The minor change point at day 4 further separates the acute phase into an initial hyperacute phase and subsequent acute components, whereas the other minor change point, at day 148, separates the convalescent phase into early and late components. This is consistent with the clinical pattern of an acute phase of illness with high mortality and a later convalescent phase associated with lower mortality risk. We also demonstrate that models with four or more inflection points (five or more pieces) do not provide substantial incremental information.

Our finding that the acute phase after an episode of AKI lasts approximately 42 days (with a 95% confidence interval of 22 to 66 days) supports the suggestion of Capuzzo et al. (18) that choosing a 28-day or 30-day mortality endpoint in AKI patients may miss a significant percentage of total disease-related mortality. At the same time, our finding is consistent with previous trial results such as Ronco et al. (14) in which most of reported mortality occurred within 35 days after initiation of dialysis. Similarly, in a study by Gastaldello et al. (16), mortality rates did not plateau until after day 50, whereas the majority of the observed mortality occurred within the first 4 weeks.

Our findings not only inform future trial design but should guide data analysis related to survival. We have demonstrated that a constant or linear hazard assumption does not hold for the survival of patients with dialysis-requiring AKI. The rate of decline in the hazard for death in the acute phase (approximately the first 6 weeks) is substantially greater than its subsequent rate of decline. Thus, survival during the acute phase of illness should be examined and modeled separately from survival beyond approximately 6 weeks.

This analysis has several strengths. The ATN study was implemented across 27 VA and non-VA clinical sites, all of whom followed a detailed study protocol for provision of intermittent hemodialysis and continuous hemodiafiltration. The study population was diverse in terms of age, race/ethnicity, geography, comorbidity, and severity of illness. We were careful to consider implications of the timing of randomization relative to midnight during the first study day to avoid misclassification of survival time when expressed in terms of days. We applied a flexible analytic approach to determine whether the mortality hazard was constant and if not (as evidenced), the day(s) on which the hazard appeared to change. As such, we did not limit ourselves to conventional dates (e.g., 28 days, 30 days, 60 days, or 90 days) as used in previous studies.

There are also several important limitations to this analysis. Because early mortality in patients with dialysis-requiring AKI is so high, the actual number of events beyond day 42 was relatively small. Thus, the power to accurately detect and estimate overall hazard and/or change point(s) beyond day 42 was limited. It is possible that repeating the analysis using a larger cohort would allow identification of a different third change point or different shape/slope of hazard considering that we used specific inclusion and exclusion criteria, such as the exclusion of patients with moderate to advanced CKD. Although the study population was diverse in terms of most demographic and clinical characteristics, these were clinical trial participants. Although we captured numerous complications associated with dialysis-requiring AKI, we did not explore whether there were distinct phases for outcomes other than mortality, such as duration of mechanical ventilation, need for surgery or other interventions, or other complications associated with AKI. Although these data may provide patients and their families with more updated, granular data on mortality risk, additional information on other AKI-related outcomes such as duration of intensive care unit stay and recovery of kidney function might also aid in clinical decision making at initiation of dialysis and other times during the course of critical illness.

In summary, using a piecewise survival model and the data from ATN study, we have established an evidence-base for analyzing survival after an episode of dialysis-requiring AKI. By so doing, we have better defined the natural history of survival after an episode of AKI—approximately half of the patients die during the hyperacute and acute phases of AKI (from the start of dialysis to day 42); another 1 in 7 die during the convalescent (early or late) phase (from day 42 to 1 year). These findings can help to inform the design of future clinical trials and the evaluation of AKI-related clinical practice.

Disclosures

P.M.P. is a Deputy Editor of CJASN.