The Application of Cure Models in the Presence of Competing Risks

A Tool for Improved Risk Communication in Population-Based Cancer Patient Survival

Abstract

Quantifying cancer patient survival from the perspective of cure is clinically relevant. However, most cure models estimate cure assuming no competing causes of death. We use a relative survival framework to demonstrate how flexible parametric cure models can be used in combination with competing-risks theory to incorporate noncancer deaths. Under a model that incorporates statistical cure, we present the probabilities that cancer patients (1) have died from their cancer, (2) have died from other causes, (3) will eventually die from their cancer, or (4) will eventually die from other causes, all as a function of time since diagnosis. We further demonstrate how conditional probabilities can be used to update the prognosis among survivors (eg, at 1 or 5 years after diagnosis) by summarizing the proportion of patients who will not die from their cancer. The proposed method is applied to Swedish population-based data for persons diagnosed with melanoma, colon cancer, or acute myeloid leukemia between 1973 and 2007.

Population-based cancer patient survival has long been used to evaluate the effectiveness of the health care system with respect to its ability to diagnose and treat cancer patients.¹ Estimates of 5-year relative survival are used extensively for this purpose in cancer registry reports, as well as in clinical research. Relative survival is defined as the ratio of all-cause survival to the survival that would be expected in the absence of cancer.² Relative survival aims to estimate “the survival so far as cancer is concerned”³—a hypothetical quantity that is today known as net survival (in the literature on population-based cancer survival) or marginal survival (in the competing-risks literature).⁴ Net survival is the survival we would observe under the counterfactual scenario in which all other causes of death are eliminated. The empirical meaning of this counterfactual is not obvious because eliminating death is a highly hypothetical intervention; we refer to Robins and Greenland⁵ for a discussion on ill-defined counterfactuals.

Recently, there has been interest in methods to estimate cancer survival in the presence of competing risks to relate the risk of dying from the cancer to that of dying from competing causes.⁶ Moreover, studies that report the probability of cure are becoming more common. Although cancer register data cannot be used to predict individual-level prospects of clinical cure, models that estimate statistical cure have been proposed.^7–12 These models partition the patients into 2 groups: those who are cured, and those who are bound to die from their cancer. The cured group will not experience excess mortality due to cancer, that is, they will eventually die from similar causes as the general population. In this setting, cure is estimated in the absence of competing risks, and, as such, cured patients are assumed to be “immortal.”^13,14

In this paper, we show how flexible parametric cure models can be used with individual-level population-based data to produce summary measures of cancer patient survival in the presence of competing risks.⁷ Specifically, we show how the following quantities can be estimated:

• The proportion of patients who will not die from the diagnosed cancer, as a function of time since diagnosis.
• Probabilities of death from causes other than the diagnosed cancer, conditional on the patients being still alive at fixed times of follow-up, for example, at 1 or 5 years after diagnosis.

Summary measures such as these can be a useful complement to existing statistical measures of survival that currently serve as a basis for risk communication.

METHODS

Data Sources

This study uses data from the Swedish Cancer Registry to which reporting of new tumors is mandatory by law.¹⁵ All cases of malignant melanoma (International Classification of Diseases, 7th Revision [ICD7]: 190x), colon cancer (ICD7: 153x), and acute myeloid leukemia (ICD7: 205.0, 205.9, 206.0, and 206.9) diagnosed in Sweden between 1973 and 2007 at ages 19–80 years were extracted from the registry. Dates of death for deceased persons up to 31 December 2012 were retrieved from the Swedish Cause of Death Register. In patients with multiple tumors of the same type, we considered only the first diagnosis. Incidental autopsy findings and benign tumors were excluded.

Statistical Analysis

Flexible parametric cure models are fitted on the log-cumulative-excess-hazard scale using restricted cubic splines to model the baseline excess hazard.⁷ The model can predict relative survival (or excess mortality) and cure. Cure is estimated by restricting the cumulative baseline-excess-hazard function to have zero slope after the last knot. The time of cure is thus defined by the user via the placement of the last knot.

We fitted flexible parametric cure models that included sex, calendar year (using restricted cubic splines), and age at diagnosis (using restricted cubic splines) where cure was assumed after 10 years. The effects of the 2 continuous variables were assumed to be time dependent and to interact.

Figure 1 illustrates the quantities of interest. The net proportion of patients bound to die from their cancer is estimated from the model by defining an asymptote to the net cumulative probability of death from cancer, that is, 1 − relative survival (Figure 1A). The cured proportion is the remaining patients (ie, 1 − net proportion bound to die from cancer).

FIGURE 1: Illustration of how the vital status of Swedish men diagnosed with colon cancer in 2007 at age 75 years can be partitioned in the absence and presence of competing causes of death. (A) Net survival, (B) crude survival, (C) crude survival further partitioned by predicted prognosis, (D) conditional probabilities of death from competing causes.

The probability of death from the diagnosed cancer at time t, estimated in the presence of other causes of deaths, is commonly referred to as a crude (in contrast to net) probability of death from the cancer, denoted here by P_{cr, can} (t), and can be calculated as

where R(u) and γ(u) are the relative survival and excess mortality, respectively, estimated from the flexible parametric cure model, whereas S^* (u) is the expected survival in the general population.^16,17 The crude proportion of patients bound to die from cancer, P_{btd, can}, was retrieved by evaluating the above integral at t = 10 years, that is, P_{btd, can} P_{cr, can} (10) cr, can because it was assumed that cure had been reached by then. The crude probability of death due to causes other than the diagnosed cancer, P_{cr, oth} (t), was calculated by substituting h^* (u) for γ(u), denoting the expected mortality rate in the general population in equation (1). Figure 1B illustrates these various quantities, showing that the crude proportion of patients bound to die from their cancer is lower than its net counterpart, as the patients are no longer assumed to be immune from competing causes of death.

Until excess mortality from the diagnosed cancer reaches zero (in the group of patients as a whole), some patients will eventually die from their cancer. In Figure 1C, patients have been partitioned into 2 groups: patients who will ultimately die from their cancer,P_{alive, can} (t), and patients who will die from other causes, P_{alive, oth} (t). The dashed line that separates the 2 groups is obtained by summing P_{btd, can} and P_{cr, oth}(t) at each time of follow-up.

It follows that P_{alive, can} (t) and P_{alive, oth} (t) can be calculated via

and

Moreover, for patients who are still alive after a specific number of years, updated prognostic information is important. Survivors have an interest in prognostic questions such as “Given that a man diagnosed with colon cancer in 2007 at age 75 years is still alive after 5 years, what is the chance that he is going to die from some cause unrelated to the diagnosed cancer?”. To this end, probabilities of death from causes other than cancer, conditional upon survival to timet, P_oth|alive(t), can be calculated via

Figure 1D shows such conditional probabilities (including 95% confidence intervals [CIs]).

Further details of the estimation of the flexible parametric cure model, software implemented, and assessment of the modeling assumptions are provided in an eAppendix (https://links.lww.com/EDE/A799).

RESULTS

Data were available for 39,895 patients with melanoma (mean age at diagnosis: 56 years), 73,921 patients with colon cancer (mean age at diagnosis: 68 years), and 8070 patients with acute myeloid leukemia (mean age at diagnosis: 63 years). Patient characteristics for each of the 3 cohorts are summarized in Table 1. The proportion of patients who died from any cause within the first 10 years of follow-up was 31% for melanoma, 66% for colon cancer, and 89% for acute myeloid leukemia.

TABLE 1: Characteristics of Patients Diagnosed with Melanoma, Colon Cancer, and Acute Myeloid Leukemia (AML) in Sweden Between 1973 and 2007

In Table 2, the estimated cure proportions for patients diagnosed at ages 50, 60, 70, and 80 years, respectively, are contrasted with the corresponding proportions of patients who are predicted to die from causes other than the diagnosed cancer, under the competing-risks model. For all 3 cancers, there is little or no difference between the 2 measures for patients aged 50 years at diagnosis. For 80-year-old men, the observed differences between the 2 measures range from 6% for colon cancer to 7% for melanoma and acute myeloid leukemia. For women, the corresponding differences are smaller. In general, women have a more favorable survival, that is, higher cure proportion and greater probability of dying from noncancer causes.

TABLE 2: Statistical Cure Proportions and Crude Proportions of Patients Who Will Die from Causes Other Than Their Cancer, Estimated Using Flexible Parametric Cure Models

Figure 2 shows the predicted 1- and 5-year outcomes (dead from cancer, dead from other causes, alive but will die from the diagnosed cancer, or alive and will die from other causes) for men diagnosed in 2007, as a function of age at diagnosis. The probability of dying from acute myeloid leukemia is strongly associated with age at diagnosis, with older patients having worse prognosis. Such a pattern is not observed for patients with melanoma or colon cancer, among whom deaths from other causes increase with advancing age. One year after diagnosis, nearly all melanoma patients who are predicted to die from their cancer are still alive, whereas for patients with colon cancer and acute myeloid leukemia (for which disease progression tends to be faster), deaths from cancer are more common in the first year. Five years after diagnosis, the majority of all patients predicted to die from acute myeloid leukemia had died.

FIGURE 2: Predicted 1- and 5-year vital status of men diagnosed with melanoma, colon cancer, or acute myeloid leukemia (AML) in Sweden in 2007, by age at diagnosis (years).

Figure 3 shows the proportion of men diagnosed in 2007 who are predicted to die from causes other than the diagnosed cancer, conditional on already having survived 1 or 5 years. For example, provided that a man diagnosed with melanoma at age 60 survives for 1 year, the probability that he will die from some cause other than melanoma is 0.85 (95% CI = 0.83–0.86). If the same man has survived 5 years, the corresponding probability is 0.95 (0.94–0.96). The 1-year conditional probabilities that men with melanoma or colon cancer will die from causes other than the diagnosed cancer are not as strongly correlated with age at diagnosis as for patients with acute myeloid leukemia. The 5-year conditional probabilities are less variable across cancer type and age at diagnosis. Moreover, because we assume that cure has been reached 10 years after diagnosis, it follows, by definition, that the probability of dying from causes unrelated to the diagnosed cancer is 1.00 for those who survive at least 10 years. Figure 4 shows how the conditional probabilities of death from causes other than cancer are affected if we relax this assumption. After pushing back the timing of cure to 11, 12, or 13 years, the estimates are very similar.

FIGURE 3: Predicted proportion of men diagnosed in 2007 with melanoma, colon cancer, or acute myeloid leukemia (AML) in Sweden who will die from other causes than their cancer, conditional upon having survived for 1 or 5 years after diagnosis, by age at diagnosis (years).

FIGURE 4: Sensitivity analysis with respect to the assumed timing of cure in relation to the diagnosis, by age at diagnosis (years). AML indicates acute myeloid leukemia.

DISCUSSION

We have proposed a statistical method that summarizes the likely prognosis of patients who have been diagnosed with a cancer for which statistical cure is a reasonable assumption. We have shown how flexible parametric cure models, used in combination with competing-risks theory, can be used to partition the probability of death from any cause into probabilities of death from various competing causes. We also showed how survivors can be subdivided into 2 groups: those who will eventually die from their cancer, and those who will die from other causes. Several other authors have proposed methods to estimate an analog to cure in a competing-risks setting. The shared feature of these models is that they estimate the proportion of patients for whom a death from some cause unrelated to the cancer is predicted to precede an eventual cancer death. Gordon¹⁸ called this quantity “personal cure” and estimated it using a latent mixture model applied to clinical trial data for breast cancer. Yu et al¹⁹ extended the work of Gordon to population-based cancer patient survival studies. They also used the term “personal cure” and adapted a mixture model for grouped survival data with competing risks previously described by Larson and Dinse.²⁰ Sasieni²¹ instead referred to this construct as “avoidance of premature death” and provided a nonparametric alternative that incorporated a period approach to estimation to achieve more accurate projections of the future prognosis of newly diagnosed patients.

We have focused on a population-based setting and use relative survival (rather than cause-specific survival) as a basis for estimation, circumventing the need for accurate cause-of-death information. The flexible parametric cure model uses restricted cubic splines for modeling the baseline excess hazard, which obviates the need to make strong assumptions about the survival-time distributions of the patient subpopulations.²² Although we must specify an appropriate number and placement of the knots used to model the baseline excess hazard, the results of flexible parametric models have been shown to be robust to the configuration of the knots.²³ Moreover, flexible parametric cure models are fitted to individual-level data. This allows the effects of important prognostic continuous variables, such as age at diagnosis, to be modeled by using splines, for example, instead of categorical analysis, leading to a higher resolution of the reported summary statistics.

We have extended the work of previous investigators by advocating reporting of conditional probabilities for patients who seek updated estimates of their prognosis. Regular updates of the prognosis, given survival for a certain period of time after diagnosis, provide critical information for cancer survivors. Updated information may decrease anxiety associated with the diagnosis of cancer and help manage the feeling of uncertainty about the future.^24,25

The timing of the cure point is crucial to our model, and the decision of where to place the last knot in the flexible parametric cure model should be based on subject-matter knowledge about the natural progression of the disease under study, and accompanied by careful sensitivity analyses.⁷ The cure point can vary substantially among cancer types and across populations. Ten years of follow-up seems to suffice for recently diagnosed patients in Sweden with melanoma, colon cancer, and acute myeloid leukemia; longer follow-up might be required for the same cancers in other countries.^26–29

Although we report results for patients diagnosed in 2007 (to ensure at least 5 years of follow-up was available), data from all patients diagnosed between 1973 and 2007 were included in the models. In scenarios in which there have been great improvements in survival over time (eg, following the introduction of new treatments), our model can easily be extended to incorporate delayed entry (ie, a period approach to estimation) to yield more accurate predictions of long-term survival.^21,30

Cure models estimated in the absence of competing risks have previously been applied to a wide selection of cancers.^31–33 Such models are useful for studies of temporal trends in patient survival or disease etiology because such models ensure that observed differences between groups of patients are not affected by differences in noncancer mortality. However, incorporating competing causes of death in the statistical analysis is important to assess the real-world risk that the patients face of dying from the diagnosed cancer, especially if the patients have reached an age where deaths from causes other than cancer are expected.

Population-based cancer patient survival studies provide a unique opportunity to assess the effect of a cancer diagnosis on all patients, in contrast to randomized clinical trials, in which the patients often represent a select group of relatively young and healthy people. Often, however, there are other limitations to what can be estimated using administrative health registers. For example, the Swedish Cancer Registry does not record treatment data, nor is there information about cancer stage for patients diagnosed before 2004. In addition, the population mortality rates used to model the probability of death from competing causes are usually stratified on sex, age, and calendar year, but not such factors as social class, lifestyle factors, or general health status—all factors strongly associated with survival. If such data are available, they should be included in the statistical model to further tailor the summary statistics toward more homogeneous groups of patients.

Given the increasing number of long-term cancer survivors, statistical summary measures that are applicable in the presence of competing causes of death are needed. The measures described in this study are of more direct interest to cancer survivors and physicians than estimates of net survival and can be a useful tool in risk communication with patients diagnosed with a curable cancer.